\section{Introduction}\label{intro}
Let $f$ be an indefinite and nondegenerate quadratic form in $n\geq 3$ real variables that is not a real multiple of a quadratic form with rational coefficients. The Oppenheim\textendash Davenport Conjecture, resolved affirmatively by G.\,A.~Margulis in \cite{Mar89}, states that every real number is an accumulation point of $f(\mathbb{Z}^n)$:~that is to say, for any $\xi \in \mathbb{R}$ and any $\varepsilon \in \mathbb{R}_{>0}$, \eq{density}{\text{there exist infinitely many }\mathbf{v}\in\mathbb{Z}^n\text{ for which } |
f(\mathbf{v}) - \xi| \leq \varepsilon.} The rich history of the Oppenheim\textendash Davenport conjecture and its seminal resolution by Margulis, among various other related topics, are extensively discussed in Margulis's survey \cite{Mar97}. The influence of Margulis's theorem and related problems continues unabated to this day. As of a few years ago, there has been a great increase of activity in proving \textit{effective} variants of Margulis's theorem for \textit{generic} quadratic forms and other homogeneous polynomials:~for instance, one often considers the $\mathrm{SL}_n(\mathbb{R})$-orbit (under the natural action) of a real homogeneous polynomial in $n$ real variables; one then has a natural notion of measure class (and thus measure-theoretic genericity) for this orbit. Let us briefly recall some recent results that exemplify this circle of problems.            


For any $\beta \in \mathbb{R}_{\geq 1}$ and any $(p, q) \in \left(\mathbb{Z}_{\geq 1}\right)^2$ with $p+q = n \geq 3,$ let $\displaystyle F_{p, q, \beta} : \mathbb{R}^n \to \mathbb{R}$ be given by \eq{generalquad}{F_{p, q, \beta}(\mathbf{x}) := \sum_{j=1}^p  \left|x_j\right|^\beta - \sum_{k=p+1}^{n} \left|x_{k}\right|^\beta.} 
\ignore{In \cite{GhoshGorodnikNevo}, Ghosh\textendash Gorodnik\textendash Nevo proved a result that is easily seen to be equivalent to the following one. \begin{thme}[{\cite[Theorem 1.2]{GhoshGorodnikNevo}}] Let $s \in (0, 1) \subset \mathbb{R}.$ Let $\xi \in \mathbb{R}$. Then for Haar-almost every $g \in \mathrm{SL}_3(\mathbb{R})$ the following holds:~for each sufficiently large $T \in \mathbb{R}_{>0}$ there exists $\mathbf{v} \in \mathbb{Z}^3$ with \[ 0 < \|\mathbf{v}\| \leq T \quad \text{and} \quad \left|-\xi + F_{2, 1, 2}(g\mathbf{v}) \right| \leq T^{-s}, \] where $\|\cdot\|$ is a certain norm on $\mathbb{R}^3.$  \end{thme} 
\noindent The proof of Ghosh\textendash Gorodnik\textendash Nevo made use of representation theory and effective mean ergodic theorems. Using comparatively elementary means (namely, first and second moment formulae for the Siegel transform in the geometry of numbers), Athreya\textendash Margulis generalized the result of  Ghosh\textendash Gorodnik\textendash Nevo in the special case $\xi = 0.$ The following theorem is easily seen to be equivalent to the one that Athreya\textendash Margulis proved in \cite{Pol}.   \begin{thme}[{\cite[Theorem 1.1]{Pol}}] 
Suppose that $n \in \mathbb{Z}_{\geq 3}$ is arbitrary. Suppose that $p$ is any integer with $1 \leq p \leq n-1.$ Let $s \in (0, n-2) \subset \mathbb{R}$ be given. Then for Haar-almost every $g \in \mathrm{SL}_n(\mathbb{R})$ the following holds:~for each sufficiently large $T \in \mathbb{R}_{>0}$ there exists $\mathbf{v} \in \mathbb{Z}^n$ with \[ 0 < \|\mathbf{v}\| \leq T \quad \text{and} \quad \left|F_{p, q, 2}(g\mathbf{v}) \right| \leq T^{-s}, \] where $\|\cdot\|$ is a certain norm on $\mathbb{R}^n.$ \end{thme} }

Generalizing earlier results of Ghosh\textendash Gorodnik\textendash Nevo \cite{GhoshGorodnikNevo} and Athreya\textendash Margulis \cite{Pol}, Kelmer\textendash Yu proved the following theorem.  

\begin{thm}[{\cite[Corollary 2]{KY2}}] \label{KY} Let $\beta \in 2\left(\mathbb{Z}_{\geq 1}\right)$, let $n \in \mathbb{Z}_{> \beta}$, and let $p$ and $q$ be elements of $\mathbb{Z}_{\geq 1}$ for which $p+q = n.$ Let $s \in (0, n - \beta) \subset \mathbb{R}.$ Let $\xi \in \mathbb{R}.$ Then for Haar-almost every $g \in \mathrm{SL}_n(\mathbb{R})$ the following holds:~for each sufficiently large $T \in \mathbb{R}_{>0}$ there exists $\mathbf{v} \in \mathbb{Z}^n$ with \[ 0 < \|\mathbf{v}\| \leq T \quad \text{and} \quad \left|-\xi + F_{p, q, \beta}(g\mathbf{v}) \right| \leq T^{-s}, \] where $\|\cdot\|$ is a certain norm on $\mathbb{R}^n.$  \end{thm} The proof of Ghosh\textendash Gorodnik\textendash Nevo made use of representation theory and effective mean ergodic theorems, while those of Athreya\textendash Margulis and then Kelmer\textendash Yu employed comparatively elementary means (namely, first and second moment formulae for the Siegel transform in the geometry of numbers).  
\ignore{The result {{\cite[Corollary 2]{KY2}}} 
 is a corollary of the powerful and explicit counting result \cite[Theorem 1]{KY2}.} 

\ignore{In \cite{GKY_a}, Ghosh\textendash Kelmer\textendash Yu establish the counting result \cite[Theorem 1.2]{GKY_a}, which allows them to conclude the following result. 

\begin{thme}[{\cite[Remark 1.5 and \S 1.3]{GKY_a}}] 
Suppose that $n \in \mathbb{Z}_{\geq 3}$ is arbitrary. Suppose that $p$ is any integer with $1 \leq p \leq n-1.$ Let $s \in (0, n-2) \subset \mathbb{R}$ be given. Let $\xi \in \mathbb{R}.$ Let ${\pmb \alpha} \in \mathbb{Q}^n.$ Then for Haar-almost every $g \in \mathrm{SL}_n(\mathbb{R})$ the following holds:~for each sufficiently large $T \in \mathbb{R}_{>0}$ there exists $\mathbf{v} \in \mathbb{Z}^n$ with \[ \|\mathbf{v}\| \leq T \quad \text{and} \quad \left| -\xi + F_{p, q, 2}\left(g\left({\pmb \alpha} + \mathbf{v}\right)\right) \right| \leq T^{-s}, \] where $\|\cdot\|$ is a certain norm on $\mathbb{R}^n.$ 
\end{thme}}      For other results of a similar nature, see {\cite{EMM, MM, LM, Bourgain, GhoshKelmer, GKY_b,  Anishnew}}. The authors previously studied problems of this sort in \cite{KS}; however, the work in \cite{KS} was limited to \textit{homogeneous} approximation, which corresponds to the special case $\xi = 0$ in the context of \equ{density}. The purpose of the present paper is to establish \textit{inhomogeneous} analogues of the results of \cite{KS}. In order both to recall the results of \cite{KS} and to present the results of the present paper, we proceed to establish some notation and definitions; much of the notation and many of the definitions that follow were first established in \cite{KS} and are recalled here for the convenience of the reader. 



\smallskip    



Now and hereafter, we shall denote by $n$ an arbitrary element of $\mathbb{Z}_{\geq 2}$ and by $\ell$ an arbitrary element of {$\mathbb{Z}_{\geq 1}$}. Elements of $\mathbb{R}^n$ and of $\mathbb{R}^\ell$ shall always be regarded as column vectors, even though they may be written as row vectors for notational convenience. If $k \in \mathbb{Z}_{\geq 1}$ is arbitrary and $E$ is any subset of $\mathbb{R}^k,$ then we define $E_{\neq 0} := E \ssm \left\lbrace\mathbf{0}_{\mathbb{R}^k}\right\rbrace.$ We say that an arbitrary point $\mathbf{v} = (v_1, \dots , v_n) \in \mathbb{Z}^n$ is \textsl{primitive} if $\gcd(v_1, \dots , v_n) = 1.$ We let $\mathbb{Z}^n_{\rm pr}$ denote the set of all primitive points of $\mathbb{Z}^n.$ We shall always denote the usual Lebesgue measure on any Euclidean space by $m$. Throughout this paper, we shall use the Vinogradov notation $\ll$ and use $\asymp$ to denote that both "$\ll$" and "$\gg$" hold; we shall attach subscripts to the symbols $\ll$ and $\asymp$ to indicate the parameters, if any, on which the implicit constants depend.         

\begin{defn}[{\cite[Definition 1.1]{KS}}]\label{order}
We define a non-strict partial order $\preccurlyeq$ on $\mathbb{R}^{\ell}$ as follows. For any $\displaystyle \mathbf{x} = (x_1, \dots , x_{\ell}) \in \mathbb{R}^{\ell}$ and any $\displaystyle \mathbf{y} = (y_1, \dots , y_{\ell}) \in \mathbb{R}^{\ell},$ we write $\mathbf{x} \preccurlyeq \mathbf{y}$ if and only if for each $j \in \{1, \dots , {\ell}\},$ we have $x_j \leq y_j.$ (Equivalently, we write $\mathbf{x} \preccurlyeq \mathbf{y}$ if and only if $\displaystyle \left(\mathbf{y} - \mathbf{x}\right) \in \left(\mathbb{R}_{\geq 0}\right)^\ell.$) \end{defn}


\begin{defn}[see {\cite[Definitions 1.2, 3.3, 3.6, and 3.7]{KS}}]\label{basicapdefns}
Let \[f = (f_1, \dots , f_{\ell}) : \mathbb{R}^n \to \mathbb{R}^{\ell} \ \ \ \ \ \text{and} \ \ \ \ \  \psi = (\psi_1, \dots , \psi_{\ell}) : \mathbb{R}_{\geq 0} \to \left(\mathbb{R}_{>0}\right)^{{\ell}}\] be given, let $\nu$ be an arbitrary norm on $\mathbb{R}^n$, and let $\mathcal{P}$ be an arbitrary subset of $\mathbb{Z}^n.$ 
\smallskip
\begin{itemize}
\item  We abuse notation and write $|f|$ to denote the function $\displaystyle (|f_1|, \dots , |f_\ell| ) :\mathbb{R}^n \to \mathbb{R}^{\ell}.$
\item  We define $\displaystyle \mathcal{Z}(f) := f^{-1}\left( \mathbf{0}_{\mathbb{R}^\ell} \right)\smallsetminus\{\mathbf{0}_{\mathbb{R}^n} \}$.     
\item We define $\displaystyle A_{f, \psi, \nu} := \left\{ \mathbf{x} \in \mathbb{R}^n : |f(\mathbf{x})| \preccurlyeq \psi\big(\nu(\mathbf{x})\big)\right\}.$
\item For any $T \in \mathbb{R}_{>0}$ and any ${\pmb\varepsilon}\in \left(\mathbb{R}_{>0}\right)^\ell,$ we define $$B_{f, {\pmb\varepsilon}, \nu, T} := \left\{  \mathbf{x} \in \mathbb{R}^n : |f(\mathbf{x})| \preccurlyeq {\pmb\varepsilon} \ \text{and} \ \nu(\mathbf{x}) \leq T \right\}.$$ 
\item We  say that $f$ is $(\psi, \nu, \mathcal{P})$-\textsl{approximable} if $A_{f, \psi, \nu} \cap \mathcal{P}$ has infinite cardinality. 
\item We  say that $f$ is \textsl{uniformly} $(\psi, \nu, \mathcal{P})$-\textsl{approximable} if {$B_{f, \psi(T), \nu, T} \cap \mathcal{P}  \neq \varnothing$ for each sufficiently large $T \in \mathbb{R}_{> 0}$}.  
\ignore{\item Let $\displaystyle t_\bullet = \left(t_k\right)_{k \in \mathbb{Z}_{\geq 1}}$ be any strictly increasing sequence of elements of $\mathbb{R}_{>0}$ with $\displaystyle \lim_{k \to \infty} t_k = \infty.$ We then say that $f$ is $\displaystyle t_\bullet$-\textsl{uniformly} $\left( \psi, \nu, \mathcal{P} \right)$-approximable if {$B_{f, \psi(t_k), \nu, t_k} \cap \mathcal{P}  \neq \varnothing$ for each sufficiently large $k \in \mathbb{Z}_{\geq 1}$}.     
\item Let $\displaystyle t_\bullet = \left(t_k\right)_{k \in \mathbb{Z}_{\geq 1}}$ be any strictly increasing sequence of elements of $\mathbb{R}_{>0}$ with $\displaystyle \lim_{k \to \infty} t_k = \infty.$ We say that $t_\bullet$ is \textsl{quasi-geometric} if, in addition to the preceding, the set $\displaystyle \left\lbrace t_{k+1}/t_k : k \in \mathbb{Z}_{\geq 1} \right\rbrace$ is bounded. 
\item We say that  $f$ is \textsl{subhomogeneous} if $f$ is Borel measurable and there exists ${\delta} = {\delta}(f) \in \mathbb{R}_{>0}$ such that for each $t \in (0, 1) \subset \mathbb{R}$ and each $\mathbf{x} \in \mathbb{R}^n$ we have $\displaystyle |f(t\mathbf{x})| \preccurlyeq t^{\left({\delta}\right)} |f(\mathbf{x})|.$}       
\item We say that  $f$ is \textsl{homogeneous} if it is Borel measurable and there exists some $\mathbf{d} = \mathbf{d}(f) = (d_1,\dots,d_\ell) \in (\mathbb{R}_{>0})^\ell$ such that for each $t \in \mathbb{R}_{\geq 0}$, each $j \in \{1, \dots , \ell\},$ and each $\mathbf{x} \in \mathbb{R}^n$ we have $\displaystyle f_j(t\mathbf{x}) = t^{d_j} f_j(\mathbf{x}).$ We refer to $\mathbf{d} = \mathbf{d}(f)$ as the \textsl{degree of} $f.$\footnote{This terminology is slightly abusive because $\mathbf{d} = \mathbf{d}(f)$ is unique if and only if each component of $f$ is nonzero.}   
\item We say that $\psi$ is \textsl{regular} if it is Borel measurable and there exist real numbers $a = a(\psi) \in \mathbb{R}_{>1}$ and $b = b(\psi) \in \mathbb{R}_{>0}$ such that for each ${z} \in \mathbb{R}_{>0}$ one has $b\psi({z}) \preccurlyeq \psi(a {z}).$
\item We say that $\psi$ is \textsl{nonincreasing} if each component function of $\psi$ is nonincreasing in the usual sense.
\end{itemize}
\end{defn}

\begin{defn}\label{shiftedfunctiondef}
For any function $f: \mathbb{R}^n \to \mathbb{R}^\ell$ and any ${\pmb \xi} \in \mathbb{R}^\ell,$ define $\displaystyle _{\pmb \xi} f: \mathbb{R}^n \to \mathbb{R}^\ell$ to be the function given by $\displaystyle _{\pmb \xi} f(\mathbf{x}) := -{\pmb \xi} + f(\mathbf{x}).$ 
\end{defn} 


\begin{defn}\label{logcorrectdefn}
For any
$s \in \mathbb{R}_{>0}$  and any $\varepsilon \in \mathbb{R}_{\geq 0},$ let $\varphi_{s, \varepsilon} : \mathbb{R}_{\geq 0} \to \mathbb{R}_{>0}$ be any regular and nonincreasing function such that for every sufficiently large $t \in \mathbb{R}_{> 1}$ we have $\varphi_{s, \varepsilon}(t)= t^{-s} \left(\log t \right)^{\varepsilon}.$
\end{defn}

\ignore{{\begin{defn}\label{logcorrectdefn}
For any $j \in \mathbb{Z}_{\geq 1},$ let $\log^{{\circ j}}$ denote the $j$-fold iterated logarithm function. For any $i \in \mathbb{Z}_{\geq 1}$, any $s \in \mathbb{R}_{>0}$, and any $\varepsilon \in \mathbb{R}_{> 0},$ let $\psi_{i, s, \varepsilon} : \mathbb{R}_{\geq 0} \to \mathbb{R}_{>0}$ be any regular and nonincreasing function such that for every sufficiently large $t \in \mathbb{R}_{> 1},$ we have \[ \psi_{i, s, \varepsilon}(t)= t^{-s} \cdot \left( \log^{{\circ(i+1)}}(t) \right)^{1 + \varepsilon} \cdot \prod_{j=1}^i \log^{{\circ j}}(t). \] For any $s \in \mathbb{R}_{>0},$ let $\psi_{s} : \mathbb{R}_{\geq 0} \to \mathbb{R}_{>0}$ be any regular and nonincreasing function such that for every $t \in \mathbb{R}_{> 1},$ we have $\psi_{s}(t) = t^{-s}.$ 
\end{defn}}}

Using our newly introduced terminology, we may now restate Theorem \ref{KY} as follows.
\begin{manualtheorem}{1.1$'$}[{\cite[Corollary 2]{KY2}}] \label{KYredux}
Let $\beta \in 2\left(\mathbb{Z}_{\geq 1}\right)$, let $n \in \mathbb{Z}_{> \beta}$, and let $p$ and $q$ be elements of $\mathbb{Z}_{\geq 1}$ for which $p+q = n.$ Let $s \in (0, n - \beta) \subset \mathbb{R}.$ Let $\xi \in \mathbb{R}.$ Let $\|\cdot\|$ be a certain norm on $\mathbb{R}^n.$ Set $f := F_{p, q, \beta}.$ Then the following hold.
\begin{itemize}   
\item[(i)] The function $\left({_{\xi}} f\right) \circ g$ is $\left(\varphi_{s,0}, \|\cdot\|, \mathbb{Z}^n_{\neq 0}\right)$-approximable for Haar-almost every $g \in \mathrm{SL}_n(\R)$.\footnote{Statement (i) is a straightforward corollary of statement (ii).}   
\item[(ii)] The function $\left({_{\xi}} f\right) \circ g$ is uniformly $\left(\varphi_{s,0}, \|\cdot\|, \mathbb{Z}^n_{\neq 0}\right)$-approximable for Haar-almost every $g \in \mathrm{SL}_n(\R)$. 
\end{itemize}
\end{manualtheorem}       


\smallskip


We now proceed to state certain results that will be formulated and proved in greater generality in \S \ref{acc}. Before we state the first result, let us recall an elementary notion from differential topology. 

\begin{defn}\label{manifoldregular} Let $M$ and $N$ be $\mathscr{C}^1$ manifolds that are Hausdorff, second-countable, and without boundary. Let $U$ be an open subset of $M.$ Let $f : M \to N$ be a function, and suppose that $f$ is continuously differentiable on $U.$ Let $x \in U$ be given. We say that $x$ is a \textsl{regular point of} $f$ if the map $\displaystyle D_x f : T_x M \to T_{f(x)} N$ is surjective.  \end{defn}


\begin{thm}\label{simplesubmerapprox}
Let $\displaystyle \psi = (\psi_1, \dots , \psi_{\ell}) : \mathbb{R}_{\geq 0} \to \left(\mathbb{R}_{>0}\right)^{\ell}$ be regular and nonincreasing. Let $\displaystyle f = (f_1, \dots , f_{\ell}) 
: \mathbb{R}^n \to \mathbb{R}^{\ell}$ be homogeneous of degree  $\mathbf{d} = \mathbf{d}(f) = (d_1,\dots,d_\ell)\in (\mathbb{R}_{>0})^\ell$. Suppose further that $f$ is continuously differentiable on $\mathbb{R}^n_{\neq 0}$, that $\mathcal{Z}(f) \neq \varnothing$, and that 
\eq{submersion}{\text{each element of $\mathcal{Z}(f)$ is a regular point of } f.} Let $\nu$ be an arbitrary norm on $\mathbb{R}^n$. Let ${\pmb \xi} \in \mathbb{R}^\ell.$ Set $d := \sum_{j=1}^\ell d_j.$ Then the following hold. 
\begin{itemize}
\item[ \rm (i)] If $\displaystyle \int_1^{\infty} t^{n-(d+1)} \left(\prod_{j=1}^\ell \psi_j(t) \right) \mathrm{d} t$ is finite $($respectively, infinite$)$,
then $\left({_{\pmb \xi}} f\right) \circ g$ is $\left(\psi, \nu, \mathbb{Z}^n_{\neq 0}\right)$-approximable for Haar-almost no $($respectively, almost every$)$ $g \in \mathrm{SL}_n(\R)$. 
\item[\rm (ii)] Suppose that $d < n$ and that the infinite series $\displaystyle \sum_{k=1}^{\infty}\left[ 2^{k(n-d)} \prod_{j=1}^\ell \psi_j\left(2^k\right) \right]^{-1}$ converges. Then $\left({_{\pmb \xi}} f\right) \circ g$ is {\sl uniformly} $\left(\psi, \nu, \mathbb{Z}^n_{\neq 0}\right)$-approximable for Haar-almost every $g \in \mathrm{SL}_n(\R)$.
\end{itemize}
\end{thm}

\begin{rmk} \rm 
Notice that no component of $f$ in the above theorem is required to be a polynomial. Notice also that neither the integral criterion in (i) nor the summatory condition in (ii) features any dependence on ${\pmb \xi} \in \mathbb{R}^\ell.$
\end{rmk}


\begin{rmk}\label{primasl} \rm    It will follow from a more general result in \S \ref{acc} that the preceding theorem remains true if one replaces   
\begin{itemize}
\item
each instance of $\mathbb{Z}^n_{\neq 0}$ by $\mathbb{Z}^n_{\rm pr}$;
\vspace{0.05in}
\item
each instance of $\mathbb{Z}^n_{\neq 0}$ by $\mathbb{Z}^n$ and each instance of $\mathrm{SL}_n(\R)$ by $\mathrm{ASL}_n(\mathbb{R})$ .
Here, $\mathrm{ASL}(\mathbb{R}) := \mathrm{SL}_n(\mathbb{R}) \ltimes \mathbb{R}^n,$ the group of affine isomorphisms of $\mathbb{R}^n$ that preserve both volume and orientation. 
\end{itemize}   \end{rmk}

\begin{rmk}\label{examples} \rm  Notice that statement (ii) of Theorem \ref{simplesubmerapprox} easily implies Theorem \ref{KY}; in fact, we may readily generalize Theorem \ref{KYredux} as follows. Let $\beta$, $p$, $q$, $n = p+q$, and $\displaystyle F_{p, q, \beta} : \mathbb{R}^n \to \mathbb{R}$ be as in \equ{generalquad}. Suppose further $1 < \beta < n.$ Since $\beta>1$, the function $\displaystyle F_{p, q, \beta}$ is continuously differentiable on $\mathbb{R}^n.$ Since $p \geq 1$ and $q \geq 1$, we have $\mathcal{Z}\left(F_{p, q, \beta}\right) \neq \varnothing.$ It is also easy to verify that each element of $\mathbb{R}^n_{\neq 0}$ is a regular point of $F_{p, q, \beta}.$ The hypotheses of Theorem \ref{simplesubmerapprox} are thus satisfied when $\ell = 1$, $d = d_1 = \beta$, and $\displaystyle f = F_{p, q, \beta}$. Now let $\xi \in \mathbb{R}$ be arbitrary, let $\nu$ be an arbitrary norm on $\mathbb{R}^n,$ and set $f = F_{p, q, \beta}$ to simplify notation. The following then hold. 
\begin{itemize}
\item The function $\displaystyle \left({_{ \xi}} f \right) \circ g$ is $\left(\varphi_{n-\beta,0}, \nu, \mathbb{Z}^n_{\rm pr}\right)$-approximable for Haar-almost every $g \in \mathrm{SL}_n(\R)$. This generalizes (i) of Theorem \ref{KYredux} to include the case of the critical exponent $s = n-\beta.$
\item The function $\displaystyle \left({_{ \xi}} f \right) \circ g$ is uniformly $\left(\varphi_{n-\beta,\varepsilon}, \nu, \mathbb{Z}^n_{\rm pr}\right)$-approximable for every $\varepsilon \in \mathbb{R}_{>1}$ and Haar-almost every $g \in \mathrm{SL}_n(\R)$. This generalizes (ii) of Theorem \ref{KYredux}. In fact, one can generalize (ii) of Theorem \ref{KYredux} even further by suitably modifying the definition of $\varphi_{n-\beta,\varepsilon}$ to include an arbitrary and finite number of iterated logarithms.  \end{itemize}  \end{rmk}     

\begin{rmk} \rm 
Our general framework also allows for easy consideration of vector-valued examples of $f.$ For instance, let $f_1 = F_{p, q, \beta}:\mathbb{R}^n \to \mathbb{R}$ as in Remark \ref{examples}, and let $f_2 : \mathbb{R}^n \to \mathbb{R}$ be an $\mathbb{R}$-linear transformation. (These functions may remind the reader of the setting in the papers \cite{Gor, Dani, BG}.) Then $f := \left(f_1, f_2\right) : \mathbb{R}^n \to \mathbb{R}^2$ satisfies the hypotheses of Theorem \ref{simplesubmerapprox} if and only if the intersection $\mathcal{Z}(f_1) \cap \mathcal{Z}(f_2)$ is nonempty and transverse. One thereby obtains a criterion for the asymptotic approximability of and a sufficient condition for the uniform approximability of almost every element in the $\mathrm{SL}_n(\R)$-orbit of ${_{ \xi}} f.$       
\end{rmk}








\ignore{In fact, Theorem \ref{simplesubmerapprox} and Remark \ref{primasl} immediately imply the following generalization of Theorem \ref{KY}. 

\begin{cor}\label{simplesubmercor}
Let $f : \mathbb{R}^n \to \mathbb{R}$ be homogeneous of degree $d = d(f) \in \mathbb{R}_{>0}.$ Suppose further that $d<n$, that $f$ is continuously differentiable on $\mathbb{R}^n_{\neq 0}$, that $\mathcal{V}(f) \neq \varnothing$, and that each element of $\mathcal{V}(f)$ is a regular point of $f.$ Let $\nu$ be an arbitrary norm on $\mathbb{R}^n$. Let $\xi \in \mathbb{R}.$ Then for Haar-almost every $g \in \mathrm{SL}_n(\R)$, the function $\displaystyle \left({_{ \xi}} f \right) \circ g$ is \begin{itemize}
\item[{\rm (i)}] $\left(\psi_{n-d}, \, \nu, \, \mathbb{Z}^n_{\rm pr}\right)$-approximable; and 
\vspace{0.05in}  
\item[{\rm (ii)}] uniformly $\left(\psi_{i, n-d, \varepsilon}, \, \nu, \, \mathbb{Z}^n_{\rm pr}\right)$-approximable for every $i \in \mathbb{Z}_{\geq 1}$ and every $\varepsilon \in \mathbb{R}_{>0}.$ 
\end{itemize} 
\end{cor}
\begin{proof} 
The desired results follow from Theorem \ref{simplesubmerapprox}, Remark \ref{primasl}, and the integral test. \end{proof}     

\begin{rmk}\label{rmkgenquad} \rm Let us discuss a collection of examples to which Theorem \ref{simplesubmerapprox} applies. Let $\beta$, $p$, $q$, $n = p+q$, and $\displaystyle F_{p, q, \beta} : \mathbb{R}^n \to \mathbb{R}$ be as in \equ{generalquad}. Suppose further $1 < \beta < n.$ Since $\beta>1$, it is easy to see that $\displaystyle F_{p, q, \beta}$ is continuously differentiable on $\mathbb{R}^n.$ Since $p \geq 1$ and $q \geq 1$, it follows that $\mathcal{V}\left(F_{p, q, \beta}\right) \neq \varnothing.$ One then easily checks by direct calculation that each element of $\mathcal{V}\left(F_{p, q, \beta}\right)$ is a regular point of $F_{p, q, \beta}.$ The hypotheses of Theorem \ref{simplesubmerapprox} are thus satisfied when $\ell = 1$, $d = d_1 = \beta$, and $\displaystyle f = F_{p, q, \beta}$. 
\end{rmk}}

 
\ignore{ 
\comm{Here are my suggestions for rewriting the remark after the theorem (the text below is just a draft):}

{Note that in the above theorem $f$ does not have to be a polynomial, only a $\mathscr{C}^1$ function, but with  an extra nonsingularity condition. 
The said condition holds for $f = F_{p, q, \beta}$, so part (ii) of Theorem \ref{simplesubmerapprox} easily implies Theorem \ref{KY} \comm{(we need to explain it! for one thing, $F_{p, q, \beta}$ is not everywhere differentiable...)}  in fact, it   implies a slight improvement involving a logarithmic correction to the critical exponent case. We also note that part (i) of Theorem \ref{simplesubmerapprox} answers in the affirmative the question of asymptotic (albeit not uniform) approximability at the critical exponent of
{Theorem \ref{KY}}.  More precisely, for any $s, \varepsilon \in \mathbb{R}_{\geq 0}$  let $\varphi_{s, \varepsilon} : \mathbb{R}_{\geq 0} \to \mathbb{R}_{>0}$ be any regular and nonincreasing function such that for every sufficiently large $x \in \mathbb{R}_{>1}$, one has $\varphi_{s, \varepsilon}(x) = x^{-s} \left(\log x \right)^{\varepsilon}.$ For $\ell=1$,    $\xi \in \mathbb{R}$, a norm $\nu$ on $\mathbb{R}^n$, $f$ as in Theorem \ref{simplesubmerapprox} with  $0 < d = d(f) < n$,  and Haar-almost every $g \in \mathrm{SL}_n(\R)$, the function $\left({_{ \xi}} f\right) \circ g$ is \begin{itemize}
\item $\left(\varphi_{n-d, 0}, \, \nu, \, \mathbb{Z}^n_{\neq 0}\right)$-approximable; and \item uniformly $\left(\varphi_{n-d, \varepsilon}, \, \nu, \, \mathbb{Z}^n_{\neq 0}\right)$-approximable for every $\varepsilon \in \mathbb{R}_{>1}.$ 
\end{itemize}}  }




We are also able to obtain results akin to Theorem \ref{simplesubmerapprox} in certain special cases wherein the nonsingularity condition \equ{submersion} does not hold. In Theorem \ref{simplesubmerapprox} and in all the aforementioned examples, the integral and summatory conditions obtained were all independent of ${\pmb \xi} \in \mathbb{R}^\ell.$ As the following two examples illustrate, this independence need no longer hold when the nonsingularity condition \equ{submersion} fails to hold.      



 

\begin{thm}\label{simpleproduct}
Let $\psi : \mathbb{R}_{\geq 0} \to \mathbb{R}_{>0}$ be  regular and nonincreasing. Let ${\omega} \in \mathbb{R}_{>0}$ be arbitrary. Let $f : \mathbb{R}^n \to \mathbb{R}$ be given by $f(\mathbf{x}) := \left(\prod_{i=1}^n \left| x_i \right| \right)^{\omega}.$ Let $\nu$ be an arbitrary norm on $\mathbb{R}^n.$ Let $\xi \in \mathbb{R}_{\geq 0}.$ Then the following hold.   
\vspace{0.05in}
\begin{itemize}
\item[ \rm (i)]  If \[ \begin{cases}
 { \displaystyle \int_1^{\infty} \frac{\psi(t)^{1/{\omega}}}{t} \left(\log t\right)^{n-2} \, \mathrm{d} t }  &  \text{ if } \xi = 0  \\ { \displaystyle \int_1^{\infty} \frac{\psi(t)}{t} \left(\log t\right)^{n-2} \, \mathrm{d} t }  &  \text{ if } \xi > 0 \end{cases} \] is finite $($respectively, infinite$)$,
then $\left({_{\xi}} f\right) \circ g$ is $\left(\psi, \nu, \mathbb{Z}^n_{\neq 0}\right)$-approximable for Haar-almost no $($respectively, almost every$)$ $g \in \mathrm{SL}_n(\R)$. \vspace{0.05in}   
\item[\rm (ii)]  If the infinite series \[ \begin{cases}
{ \displaystyle \sum_{k=1}^{\infty}\left[ k^{n-1} \psi\left(2^k\right)^{1/{\omega}} \right]^{-1} }   &  \text{ if } \xi = 0  \\ { \displaystyle \sum_{k=1}^{\infty}\left[ k^{n-1} \psi\left(2^k\right) \right]^{-1} }  &  \text{ if } \xi > 0 \end{cases} \]  converges, then $\left({_{\xi}} f\right) \circ g$ is {\sl uniformly} $\left(\psi, \nu, \mathbb{Z}^n_{\neq 0}\right)$-approximable for Haar-almost every $g \in \mathrm{SL}_n(\R)$.
\end{itemize}
\end{thm}  


\smallskip


\ignore{\smallskip

\noindent Let $\mathcal{Q}$ denote the set of all $q \in \mathbb{Q}_{>0}$ for which there exist $q_1, q_2 \in \mathbb{Z}_{\geq 1}$ such that $q_1$ is odd, $q_2$ is odd, and $q = q_1/q_2.$ We then have the following variant of the previous result.    

\begin{thm}\label{simplenobarsproduct}
Let $\psi : \mathbb{R}_{\geq 0} \to \mathbb{R}_{>0}$ be  regular and nonincreasing, and suppose that $\displaystyle \lim_{t \to \infty} \psi(t) = 0.$ Let $q \in \mathcal{Q}$ be arbitrary. Let $f : \mathbb{R}^n \to \mathbb{R}$ be given by $f(\mathbf{x}) := \left(\prod_{j=1}^n x_j \right)^q.$ Let $\nu$ be an arbitrary norm on $\mathbb{R}^n.$ Let $\xi \in \mathbb{R}.$ Then the following hold.   
\vspace{0.05in}
\begin{itemize}
\item[ \rm (i)]  If \[ \begin{cases}
 { \displaystyle \int_1^{\infty} \frac{(\psi(t))^{1/q}}{t} \left(\log t\right)^{n-2} \, \mathrm{d} t }  &  \text{ if } \xi = 0  \\ { \displaystyle \int_1^{\infty} \frac{\psi(t)}{t} \left(\log t\right)^{n-2} \, \mathrm{d} t }  &  \text{ if } \xi \neq 0 \end{cases} \] is finite $($respectively, infinite$)$,
then $\left({_{\xi}} f\right) \circ g$ is $\left(\psi, \nu, \mathbb{Z}^n_{\neq 0}\right)$-approximable for Haar-almost no $($respectively, almost every$)$ $g \in \mathrm{SL}_n(\R)$. \vspace{0.05in}   
\item[\rm (ii)]  If the infinite series \[ \begin{cases}
{ \displaystyle \sum_{k=1}^{\infty}\left[ k^{n-1} \left( \psi\left(2^k\right) \right)^{1/q} \right]^{-1} }   &  \text{ if } \xi = 0  \\ { \displaystyle \sum_{k=1}^{\infty}\left[ k^{n-1} \psi\left(2^k\right) \right]^{-1} }  &  \text{ if } \xi \neq 0 \end{cases} \]  converges, then $\left({_{\xi}} f\right) \circ g$ is {\sl uniformly} $\left(\psi, \nu, \mathbb{Z}^n_{\neq 0}\right)$-approximable for Haar-almost every $g \in \mathrm{SL}_n(\R)$.
\end{itemize}
\end{thm}

\noindent  Our next result is of interest because of its relation to the Khintchine\textendash Groshev
Theorem. }



\begin{thm}\label{simplemax}
Let $\psi : \mathbb{R}_{\geq 0} \to \mathbb{R}_{>0}$ be  regular and nonincreasing. Let $p \in \{1, \dots , n - 1 \}$ and $\mathbf{z} = (z_1, \dots , z_p) \in \left(\mathbb{R}_{>0}\right)^p$ be given. Let $f: \mathbb{R}^n \to \mathbb{R}$ be given by \[ f(x_1, \dots , x_n) := \max\left\lbrace |x_i|^{z_i}: 1 \leq i \leq p\right\rbrace. \] Set $z := \sum_{i=1}^p  z_i^{-1}.$ Let $\nu$ be an arbitrary norm on $\mathbb{R}^n.$ Let $\xi \in \mathbb{R}_{\geq 0}.$ Then the following hold.   
\vspace{0.05in}       
\begin{itemize}
\item[ \rm (i)] If \[ \begin{cases}
 { \displaystyle \int_1^{\infty} \psi(t)^z  {t}^{n - (p+1)} \ \mathrm{d} t  }  &  \text{ if } \xi = 0  \\ { \displaystyle \int_1^{\infty} \psi(t) \, {t}^{n - (p+1)} \ \mathrm{d} t  }  &  \text{ if } \xi > 0 \end{cases} \] is finite $($respectively, infinite$)$,
then $\left({_{\xi}} f\right) \circ g$ is $\left(\psi, \nu, \mathbb{Z}^n_{\neq 0}\right)$-approximable for Haar-almost no $($respectively, almost every$)$ $g \in \mathrm{SL}_n(\R)$. \vspace{0.05in}
\item[\rm (ii)] If the infinite series \[ \begin{cases}
{ \displaystyle \sum_{k = 1}^\infty 
\left[ 2^{(n-p)k} \psi\left(2^k\right) ^z \right]^{-1}}   &  \text{ if } \xi = 0  \\ { \displaystyle \sum_{k = 1}^\infty 
\left[ 2^{(n-p)k} \, \psi\left(2^k\right) \right]^{-1}}  &  \text{ if } \xi > 0 \end{cases} \] converges, then $\left({_{\xi}} f\right) \circ g$ is {\sl uniformly} $\left(\psi, \nu, \mathbb{Z}^n_{\neq 0}\right)$-approximable for Haar-almost every $g \in \mathrm{SL}_n(\R)$.
\end{itemize}
\end{thm}  

\ignore{  \begin{rmk} \rm 
\begin{itemize}
\item[]
\item[(i)] Let $q$ be any element of $\mathbb{Q}$ for which there exist $q_1, q_2 \in \mathbb{Z}_{\geq 1}$ such that $q = q_1/q_2$, $\gcd(q_1, q_2) = 1$, and $\gcd(2, q_2) = 1.$ Let $f : \mathbb{R}^n \to \mathbb{R}$ be given by $f(\mathbf{x}) := \left(\prod_{j=1}^n  x_j \right)^q.$

\end{itemize}

\end{rmk} }    

\begin{rmk}\label{singularprimasl} \rm
The assertions in Remark \ref{primasl} apply to Theorems \ref{simpleproduct} 
and \ref{simplemax} as well. As in Remark \ref{examples}, one may easily deduce corollaries of the preceding two theorems for $\psi$ of the form $\varphi_{s,\varepsilon}.$     \end{rmk}   



\noindent Theorems \ref{simplesubmerapprox}, \ref{simpleproduct}, and \ref{simplemax} are immediate special cases of Theorems \ref{submerapprox}, \ref{productcoord}, and \ref{khintchine}; the latter three theorems are consequences (albeit not immediate ones) of Theorems \ref{asymptoticmain} and \ref{mainuniformresult}. (Likewise, the assertions in Remark \ref{primasl} are immediate consequences of Theorem \ref{submerapprox}.) Results similar to the aforementioned ones were established by the authors in \cite{KS} for the special case ${\pmb \xi } = \mathbf{0}_{\mathbb{R}^\ell}$: see \cite[Theorems 3.4 and 3.8]{KS}. 

\begin{rmk}\label{simplezerovsnonzero} \rm  
From the preceding two theorems, we see that the failure of $f : \mathbb{R}^n \to \mathbb{R}$ to satisfy the nonsingularity condition \equ{submersion} of Theorem \ref{simplesubmerapprox} is necessary, but not sufficient, to ensure dependence on $\xi.$ This is an interesting phenomenon that is worthy of further investigation. It would also be interesting to understand whether there exist reasonably well-behaved $f: \mathbb{R}^n \to \mathbb{R}^\ell$ that exhibit dependence on ${\pmb \xi}$ finer than the rather crude "zero or nonzero" dependence.    \end{rmk}




\ignore{\comm{Also two more suggestions:
\begin{itemize}
\item we spend so much time in the proof dealing with the ASL case, that it is strange that we never mention what it actually means. Maybe explain it somewhere?
\item Same about $\ell > 1$. Maybe have an example where $f$ is a pair of quadratic forms, or a pair (quadratic, linear)?
\end{itemize} }

\comm{MISHEL: I tried to address the first bullet point above in Remarks \ref{primasl} and \ref{singularprimasl}. Now that the submersion approximation theorem is much simpler (no restriction to the sphere), does that address the suggestion in the second bullet point? Or should we say something like the gradients should be nonzero and not scalar multiples of each other for $\ell=2.$ But I think that's pretty obvious that it's an easily checkable criterion for $\ell=2.$ Finally, I thought it might be good to put the definition of subhomogeneity between Lemmata 2.1 and 2.2, so that's what I ended up doing.  }}


\section{General results}\label{gen}

{We proceed by laying the groundwork necessary to state and prove our general results. Many of the definitions that follow were first introduced by the authors in \cite{KS}.   }


\smallskip

Let $\mathrm{ASL}(\mathbb{R}) := \mathrm{SL}_n(\mathbb{R}) \ltimes \mathbb{R}^n.$ We assume that $\mathrm{ASL}_n(\mathbb{R})$ and each subgroup thereof act on $\mathbb{R}^n$ in the usual manner: for any $\langle h, \mathbf{z} \rangle \in \mathrm{ASL}_n(\mathbb{R})$ and any $\mathbf{x} \in \mathbb{R}^n,$ we have $\langle h, \mathbf{z} \rangle \mathbf{x} = \mathbf{z} + h\mathbf{x}$. Let $G$ be a closed subgroup of $\mathrm{ASL}_n(\mathbb{R})$ and let $\Gamma$ denote the subgroup of $G$ given by \eq{gamma}{ \Gamma := \{ g \in G : g\mathbb{Z}^n = \mathbb{Z}^n \}.} Now and hereafter, we assume that $\Gamma$ is a lattice in $G$. (Each particular example of a closed subgroup $G \leq \mathrm{ASL}_n(\mathbb{R})$ that we shall consider will indeed satisfy this condition.) Set $X := G/\Gamma.$ We let $\mu_G$ denote the Haar measure on the unimodular group $G$ that is normalized so that $\mathrm{vol}\left(G/\Gamma\right) = 1.$ We then let $\mu_X$ denote the unique $G$-invariant Radon probability measure on $X.$ 

\smallskip   

We may identify $X$ and $\{g\mathbb{Z}^n : g \in G\}$ via the bijection $g\Gamma \longleftrightarrow g\mathbb{Z}^n$; we then equip $\{g\mathbb{Z}^n : g \in G\}$ with the quotient topology of $X$ by declaring the aforementioned bijection to be a homeomorphism. Now let ${\mathcal{P}}$ be any $\Gamma$-invariant subset of $\mathbb{Z}^n.$ Given any $\Lambda \in X,$ fix any $g \in G$ for which $\Lambda = g\mathbb{Z}^n$; then define $\Lambda_{{\mathcal{P}}} := g{{\mathcal{P}}}.$ Then $\Lambda_{{\mathcal{P}}}$ is well-defined because ${{\mathcal{P}}}$ is $\Gamma$-invariant. Given any function $f : \mathbb{R}^n \to \mathbb{R}_{\geq 0},$ we define its ${{\mathcal{P}}}$-\textsl{Siegel transform} $\displaystyle \widehat{f}^{{\,}^{\mathcal{P}}} : X \to {[0, \infty]}$ by \[ \widehat{f}^{{\,}^{\mathcal{P}}}(\Lambda) := \sum_{\mathbf{v} \in \Lambda_{{\mathcal{P}}}} f(\mathbf{v}). \]
The Siegel and Rogers axioms are as follows. 


\begin{defn}[{\cite[Definition 2.1]{KS}}]\label{SiegelRogers} Let $G$ and ${{\mathcal{P}}}$ be as above.
\smallskip

\item[ \rm (i)] We say that $G$ is of ${{\mathcal{P}}}$-\textsl{Siegel type} if there exists a constant $c = {c}_{{{\mathcal{P}}}} \in \mathbb{R}_{>0}$ such that for any bounded and compactly supported Borel measurable function $f: \mathbb{R}^n \to \mathbb{R}_{\geq 0}$ we have \[ \int_{X}\widehat{f}^{{\,}^{{\mathcal{P}}}} \, \mathrm{d}\mu_X = c \int_{\mathbb{R}^n}f \, \mathrm{d} m.\] 

\smallskip

\item[ \rm (ii)]  
Let $\displaystyle r \in \mathbb{R}_{\geq 1}$ be given. We say that $G$ is of $\displaystyle \left({{\mathcal{P}}}, r \right)$-\textsl{Rogers type} if there exists a constant $D = D_{\mathcal{P}, r} \in \mathbb{R}_{>0}$ such that for any bounded Borel $E \subset \mathbb{R}^n$ with $m(E) > 0$ we have \[\norm{\widehat{\mathbbm{1}_E}^{{}_{{\mathcal{P}}}} - \left(\int_{X}\widehat{\mathbbm{1}_E}^{{}_{{\mathcal{P}}}} \,d\mu_X \right) \mathbbm{1}_X}_{r} \leq D \cdot 
m(E)^{1/r}. \]  \end{defn}




There are various interesting examples of $(G, \Gamma, \mathcal{P})$ that satisfy both conditions (i) (Siegel) and (ii) (Rogers) of Definition \ref{SiegelRogers}:~see \cite[Theorems 2.5, 2.6, and 2.8]{KS} and the references therein. These axioms are expedient because they can be used in tandem with the Borel\textendash Cantelli Lemma (for the purpose of uniform approximation) and to prove analogues of W.\,M.~Schmidt's famous counting result \cite[Theorems 1 and 2]{Schmidt} (for the purpose of asymptotic approximation). 
We now recall a result from \cite{KS} that records the relevant consequences of the Siegel and Rogers axioms.  

\begin{thm}\label{genericcounting} \cite[Theorem 2.9]{KS}
Let $G$ be a closed subgroup of ${\rm ASL}_n(\mathbb{R})$, let $\Gamma$ be as in \equ{gamma}, and let ${{\mathcal{P}}}$ be a $\Gamma$-invariant subset of $\mathbb{Z}^n.$ Suppose $G$ is of $\mathcal{P}$-Siegel type with $c=c_{\mathcal{P}}$. Let $E$ be a Borel measurable subset of $\mathbb{R}^n.$ 
\smallskip  
\begin{itemize}
\item[\rm (i)] If $m(E) < \infty,$ then $\displaystyle \mu_X\left(\left\lbrace \Lambda \in X: \#\left(\Lambda_{{\mathcal{P}}} \cap E\right) < \infty \right\rbrace\right) = 1.$  
\end{itemize}
\smallskip
For the remaining statements of this theorem, suppose in addition to the preceding hypotheses that we are given $r \in \mathbb{R}_{> 1}$ for which $G$ is of $\displaystyle \left(\mathcal{P}, r \right)$-Rogers type. 
\smallskip
\begin{itemize}
\item[\rm (ii)] Suppose $m(E) = \infty$. Let $\|\cdot\|$ be a norm on $\mathbb{R}^n$. Then for $\mu_X$-almost every $\Lambda \in X,$ we have  \eq{limit}{\lim_{t \to \infty } \frac{\# \left\lbrace\mathbf{x}\in\left(\Lambda_{\mathcal{P}} \cap E\right) : \|\mathbf{x}\| \leq t \right\rbrace}{c\, m\left(\{\mathbf{x}\in  E: \|\mathbf{x}\| \leq t \}\right)} = 1.} 
\smallskip
In particular,  $\displaystyle \mu_X\left(\left\lbrace \Lambda \in X: \#\left(\Lambda_{{\mathcal{P}}} \cap E\right) = \infty \right\rbrace\right) = 1.$   
\smallskip  
\item[\rm (iii)] Let $\displaystyle\{F_k\}_{k \in \mathbb{Z}_{\geq 1}}$ be a collection of 
Borel measurable subsets of $\mathbb{R}^n$ with $0 < m(F_k) < \infty$ for each $k \in \mathbb{Z}_{\geq 1}$. Suppose $\displaystyle \sum_{k=1}^{\infty} m(F_k)^{1-r} < \infty.$ Then the following holds: for $\mu_X$-almost every $\Lambda \in X,$ there exists some $k_\Lambda \in \mathbb{Z}_{\geq 1}$ such that for each integer $k \geq k_\Lambda,$ we have $\displaystyle \Lambda_{\mathcal{P}} \cap F_k \neq \varnothing.$ 
\end{itemize}           
\end{thm}  


\smallskip

{In the authors' previous paper \cite{KS}, the assumption on the function $f: \mathbb{R}^n \to \mathbb{R}^\ell$ whose values near zero were being approximated was \textit{subhomogeneity}; we recall here the definition of this property. }  
\begin{defn}\label{subhom} Let $f: \mathbb{R}^n \to \mathbb{R}^\ell.$ We say that $f$ is \textsl{subhomogeneous} if it is Borel measurable and there exists ${\delta} = {\delta}(f) \in \mathbb{R}_{>0}$ such that for each $t \in (0, 1) \subset \mathbb{R}$ and each $\mathbf{x} \in \mathbb{R}^n$ we have $\displaystyle |f(t\mathbf{x})| \preccurlyeq t^{{\delta}} |f(\mathbf{x})|.$ \end{defn}  
\noindent In this paper, we wish to prove analogues of \cite[Theorems 3.4 and 3.8]{KS} in which the subhomogeneous function $f : \mathbb{R}^n \to \mathbb{R}^\ell$ therein is replaced by a function of the form $_{\pmb \xi} F,$ where $F : \mathbb{R}^n \to \mathbb{R}^\ell$ is some sufficiently well-behaved Borel measurable function and ${\pmb \xi} \in \mathbb{R}^\ell$. The technical starting point for doing so is noting that the subhomogeneity assumption on $f$ in \cite{KS} was needed only in order to invoke the conclusions of two important lemmata from \cite{KS}; we now recall the statements of these lemmata for the convenience of the reader. 
\begin{lem}\cite[Lemmata 3.1 and 3.5]{KS}\label{3.1and3.5}
Let $\displaystyle f
: \mathbb{R}^n \to \mathbb{R}^{\ell}$ be subhomogeneous, and let $\displaystyle {\delta} = {\delta}_f \in \mathbb{R}_{>0}$ as in Definition \ref{subhom}. Let $\eta$ and $\nu$ be arbitrary norms on $\mathbb{R}^n$. The following then hold. 
\begin{itemize} 
\item[\rm (i)] Let $s \in \mathbb{R}_{>0}$. Then $m(A_{f, s\psi, \eta}) < \infty$ if and only if $m(A_{f, \psi, \nu}) < \infty$.
\item[\rm (ii)] Let  $t \in (0, 1),$ $T \in \mathbb{R}_{>0},$ and ${\pmb\varepsilon} \in \left(\mathbb{R}_{>0}\right)^\ell.$ Then $\displaystyle {t} B_{f, {\pmb\varepsilon}, \nu, T} \subseteq B_{f, {t}^{{\delta}}{\pmb\varepsilon}, \nu, {t} T}$.   
\item[\rm (iii)] {There exists $\displaystyle C^* = C_{\nu, \eta}^* \in \mathbb{R}_{\geq 1}$ such that for each $\displaystyle C \in \mathbb{R}_{\geq {C^*}},$ each $T \in \mathbb{R}_{>0},$ and each ${\pmb\varepsilon} \in \left(\mathbb{R}_{>0}\right)^\ell,$ we have $\displaystyle B_{f, {\pmb\varepsilon}, \nu, T} \subseteq C \, B_{f, C^{-{\delta}}{\pmb\varepsilon}, \eta, T}$ and $\displaystyle B_{f, {\pmb\varepsilon}, \eta, T} \subseteq C \, B_{f, C^{-{\delta}}{\pmb\varepsilon}, \nu, T} $. } 
\end{itemize}
\end{lem}    




\noindent In essence, the above lemma shows that the subhomogeneity of $f$ is sufficient to ensure that the volumes of the sets $A_{f, \psi, \nu}$ and $B_{f, {\pmb\varepsilon}, \nu, T}$ are well-behaved under change-of-norms and under scaling of any of the arguments $\psi, {\pmb\varepsilon}, T$ by arbitrary elements of $\mathbb{R}_{>0}.$  Let us also recall another lemma from \cite{KS} to which we shall need to refer. 


\begin{lem}\cite[Lemma 3.2]{KS} \label{KSLemma3.2}
Let $\displaystyle \psi = \left(\psi_1, \dots , \psi_{\ell}\right): \mathbb{R}_{\geq 0} \to \left(\mathbb{R}_{>0}\right)^{\ell}$ be regular and nonincreasing. Then the following holds: for any $c \in \mathbb{R}_{\geq 0}$ there exists $s \in \mathbb{R}_{>0}$ such that for each $x \in [0, c]$ and each $y \in \mathbb{R}_{>c},$ one has $\displaystyle \psi(y-x) \preccurlyeq s\psi(y).$
\end{lem}






We now  formulate definitions that axiomatize   {(i) of Lemma \ref{3.1and3.5} and certain desirable properties furnished by (ii) and (iii) of the same lemma.  } 


\ignore{ \rmv{~see Definitions \ref{asympmeasureinvariancedefn} and \ref{unifmeasureinvariancedefn}. The analogues of \cite[Theorems 3.4 and 3.8]{KS} then follow immediately:~see Theorems \ref{asymptoticmain} and \ref{mainuniformresult}.  We then prove a volume estimate in Theorem \ref{submersionvol} that allows us to establish a generalization of Theorem \ref{simplesubmerapprox}:~see Theorem \ref{submerapprox}. Lastly, we consider two examples that, due to the presence of singularities, are not within the purview of Theorems \ref{submersionvol} and \ref{submerapprox}:~see Theorems \ref{productcoord} and \ref{khintchine}. }            \comm{Let us remove this description for now, we'll put it back later.}}




\begin{defn}\label{asympmeasureinvariancedefn} Let $f : \mathbb{R}^n \to \mathbb{R}^{\ell}$ be Borel measurable, and let $\psi : \mathbb{R}_{\geq 0} \to \left(\mathbb{R}_{>0}\right)^\ell$ be regular and nonincreasing. 
We say that the pair $(f, \psi)$ is \textsl{asymptotically acceptable} if the following holds:~for any $s \in \mathbb{R}_{>0}$ and any norms $\nu_1, \nu_2$ on $\mathbb{R}^n,$ we have \eq{asympproperty}{ m\left( A_{f, \psi, \nu_1} \right) < \infty \quad \text{if and only if} \quad m\left( A_{f, s\psi, \nu_2} \right) < \infty. } 

\end{defn}


\begin{defn}\label{unifmeasureinvariancedefn}
Let $f : \mathbb{R}^n \to \mathbb{R}^{\ell}$ and $\psi : \mathbb{R}_{\geq 0} \to \left(\mathbb{R}_{>0}\right)^\ell$ each be Borel measurable. {We say that the pair $(f, \psi)$ is \textsl{uniformly acceptable} if the following holds:~for any $s_1, s_2, s_3, s_4 \in \mathbb{R}_{>0}$ and any norms $\nu_1, \nu_2$ on $\mathbb{R}^n,$  we have 
\eq{unifproperty}{ 0 < \liminf_{T \to \infty} \frac{m\left(B_{f, s_3\psi(T), \nu_2, s_4 T}\right)}{m\left(B_{f, s_1\psi(T), \nu_1, s_2 T}\right)} \leq \limsup_{T \to \infty} \frac{m\left(B_{f, s_3\psi(T), \nu_2, s_4 T}\right)}{m\left(B_{f, s_1\psi(T), \nu_1, s_2 T}\right)} < \infty.}   } \end{defn}


\smallskip  


\begin{rmk} \rm   
Note that if $f: \mathbb{R}^n \to \mathbb{R}^\ell$ is subhomogeneous and $\psi : \mathbb{R}_{\geq 0} \to \left(\mathbb{R}_{>0}\right)^{\ell}$ is regular and nonincreasing, then Lemma \ref{3.1and3.5} implies that the pair $(f, \psi)$ is both asymptotically acceptable and uniformly acceptable. \ignore{Also it is easy to see that the regularity and monotonicity of $\psi$ make it possible to simplify the above definitions. Namely, to prove that $(f,\psi)$ is asymptotically acceptable it is enough to check \equ{asympproperty} for $\nu_1 = \nu_2 = \nu$, some fixed norm on $\mathbb{R}^n$; the same goes about uniform acceptability and \equ{unifproperty}.  For now I just added it to the remark, but perhaps it will be better to state this observation  in more precise terms, turn it into a lemma and prove it, since it plays a significant role in our proofs.} \end{rmk}    

\noindent Let us now state and prove a lemma that will simplify various proofs in \S 3 that concern the verification of asymptotic and uniform acceptability. 

\begin{lem}\label{scalingonly} 
Let $f : \mathbb{R}^n \to \mathbb{R}^{\ell}$ and $\psi : \mathbb{R}_{\geq 0} \to \left(\mathbb{R}_{>0}\right)^\ell$ each be Borel measurable. 
\begin{itemize}
\item[{\rm (i)}] Suppose that $\psi$ is regular and nonincreasing. Suppose that there exists some norm $\nu$ on $\mathbb{R}^n$ such that for any $s \in \mathbb{R}_{>0}$ we have \[ m\left( A_{f, \psi, \nu} \right) < \infty \quad \text{if and only if} \quad m\left( A_{f, s\psi, \nu} \right) < \infty. \] Then the pair $(f,\psi)$ is asymptotically acceptable.    

\vspace{0.05in}

\item[{\rm (ii)}] Suppose that there exists some norm $\nu$ on $\mathbb{R}^n$ such that for any $s_1, s_2, s_3, s_4 \in \mathbb{R}_{>0}$ we have \[ 0 < \liminf_{T \to \infty} \frac{m\left(B_{f, s_3\psi(T), \nu, s_4 T}\right)}{m\left(B_{f, s_1\psi(T), \nu, s_2 T}\right)} \leq \limsup_{T \to \infty} \frac{m\left(B_{f, s_3\psi(T), \nu, s_4 T}\right)}{m\left(B_{f, s_1\psi(T), \nu, s_2 T}\right)} < \infty. \] Then the pair $(f,\psi)$ is uniformly acceptable.   
\end{itemize}
\end{lem}   
\begin{proof}
Let $\eta$ be an arbitrary norm on $\mathbb{R}^n.$ 
\begin{itemize}
\item[(i)] Let $a = a(\psi) \in \mathbb{R}_{>1}$ and $b = b(\psi) \in \mathbb{R}_{>0}$ be as in Definition \ref{basicapdefns}. Let $\nu$ be a norm on $\mathbb{R}^n$ as in the hypotheses of (i). Fix $k \in \mathbb{Z}_{\geq 1}$ for which $\displaystyle a^{-k} \, \nu \leq \eta \leq a^k \, \nu.$ For every $\mathbf{x} \in \mathbb{R}^n,$ we have \[ \psi\big(\eta(\mathbf{x})\big) \preccurlyeq \psi\big(a^{-k} \, \nu(\mathbf{x}) \big) \preccurlyeq b^{-k} \, \psi\big(\nu(\mathbf{x})\big). \] For every $\mathbf{x} \in \mathbb{R}^n,$ we similarly have \[ b^k \, \psi\big(\nu(\mathbf{x})\big) \preccurlyeq \psi\big(\eta(\mathbf{x})\big). \] It follows that for each $s \in \mathbb{R}_{>0},$ we have \[ m\left( A_{f, {b^k s \psi}, \nu} \right) \leq m\left( A_{f, {s\psi}, \eta} \right) \leq m\left( A_{f, {b^{-k} s \psi}, \nu} \right). \]
{This implies the desired result.}

\vspace{0.05in}

\item[(ii)] We do not assume that $\psi$ is regular, and we do not assume that $\psi$ is nonincreasing. Let $\nu$ be a norm on $\mathbb{R}^n$ as in the hypotheses of (ii). Fix $C \in \mathbb{R}_{>1}$ for which $\displaystyle C^{-1} \, \nu \leq \eta \leq C \, \nu.$ Let $s_1, s_2, T \in \mathbb{R}_{>0}$ be arbitrary. We clearly have \[ m\left(B_{f, s_1\psi(T), \nu, {C^{-1} s_2 T}}\right) \leq m\left(B_{f, s_1\psi(T), \eta, s_2 T}\right) \leq m\left(B_{f, s_1\psi(T), \nu, {C s_2 T}}\right). \] The desired result now follows.  \end{itemize} \end{proof}


We now proceed to establish our main theorems.  

\smallskip

\begin{thm}\label{asymptoticmain} Let $G$ be a closed subgroup of ${ \rm ASL}_n(\mathbb{R})$, let $\Gamma$ be as in $\equ{gamma},$ and let ${{\mathcal{P}}}$ be a $\Gamma$-invariant subset of $\mathbb{Z}^n.$ Suppose $G$ is of $\mathcal{P}$-Siegel type. Let $\displaystyle {c} = {c}_{{{\mathcal{P}}}}$ be as in Definition {\rm\ref{SiegelRogers}(i)}. Let $f : \mathbb{R}^n \to \mathbb{R}^{\ell}$ be Borel measurable. Let $\psi : \mathbb{R}_{\geq 0} \to \left(\mathbb{R}_{>0}\right)^\ell$ be regular and nonincreasing. 
Suppose that the pair $(f, \psi)$ is asymptotically acceptable. Let $\eta$ and $\nu$ be arbitrary norms on $\mathbb{R}^n.$ 
\vspace{0.055in}  
\begin{itemize}
\item[\rm (i)] Suppose $\displaystyle m\left( A_{f, \psi, \eta} \right) < \infty.$ Then for almost every $g \in G$ the function $f\circ g$ is not $(\psi, \nu, \mathcal{P})$-approximable.   
\smallskip

\item[\rm (ii)] Suppose $\displaystyle m\left( A_{f, \psi, \eta} \right) = \infty,$  and suppose that we are given $r \in \mathbb{R}_{> 1}$  for which $G$ is of $\displaystyle \left(\mathcal{P}, r\right)$-Rogers type.  Then \ignore{for each nonempty compact subset $K$ of $G$  there exist constants $D_K \in \mathbb{R}_{\geq 1},$ $E_K \in \mathbb{R}_{\geq 0}$  and $\displaystyle J_K \in \mathbb{R}_{\geq 1}$ such that for $\mu_G$-almost every $g \in K$ we have \begin{align}\label{limsup}
\limsup_{T \to \infty}\frac{{\#}\left\lbrace \mathbf{v} \in {{\mathcal{P}}} : \left|( f \circ  g )(\mathbf{v})\right| \preccurlyeq \psi\big( \nu(\mathbf{v}) \big) \ \mathrm{and} \ 2 D_K E_K < \nu(\mathbf{v}) \leq T \right\rbrace}{m\left(\left\lbrace \mathbf{t} \in \mathbb{R}^n : \left| f(\mathbf{t}) \right| \preccurlyeq J_K \psi\big(\nu(\mathbf{t})\big) \ \mathrm{and} \ {E_K} < \nu(\mathbf{t}) \leq D_K T + E_K \right\rbrace\right)} \leq {{c}_{{\mathcal{P}}}} \qquad \end{align} and 
\begin{align}\label{liminf} \ \quad 
\liminf_{T \to \infty} \frac{{\#}\left\lbrace \mathbf{v} \in {{\mathcal{P}}} : \left|( f \circ  g )(\mathbf{v})\right| \preccurlyeq \psi\big( \nu(\mathbf{v})\big) \ \mathrm{and} \ {E_K D_K^{-1}} < \nu(\mathbf{v}) \leq D_K E_K + D_K T \right\rbrace}{m\left(\left\lbrace \mathbf{t} \in \mathbb{R}^n : \left| f(\mathbf{t}) \right|  \preccurlyeq J_K^{-1}\psi\big((\nu(\mathbf{t})\big) \ \mathrm{and} \ 2E_K < \nu(\mathbf{t}) \leq T \right\rbrace\right)} \geq {{c}_{{\mathcal{P}}}}. \end{align} 



\smallskip



\noindent   If $K \subseteq \mathrm{SL}_n(\R),$ then each of the inequalities in \eqref{limsup} and \eqref{liminf} holds with $E_K = 0.$ 


\noindent   In particular,}for almost every $g \in G$ the function $f \circ g$ is $(\psi, \nu, {{\mathcal{P}}})$-approximable. \end{itemize} \end{thm}

\begin{proof} 
Let $a = a(\psi) \in \mathbb{R}_{>1}$ and $b = b(\psi) \in \mathbb{R}_{>0}$ be as in Definition \ref{basicapdefns}. Let us denote elements of ${\rm ASL}_n(\mathbb{R})$ by $\langle h,\mathbf{z} \rangle$, where $h \in \mathrm{SL}_n(\R)$ and $\mathbf{z} \in \mathbb{R}^n$; that is,
$\langle h,\mathbf{z}\rangle : \mathbb{R}^n \to \mathbb{R}^n$ is the affine map given by $\mathbf{x} \mapsto \mathbf{z} + h\mathbf{x}$. For any $h \in \mathrm{SL}_n(\R)$, let $\|h\|$ denote the operator norm of $h$ that is given by \eq{norm}{\displaystyle \|h\|:= \sup\left\lbrace \nu(h\mathbf{x}) : \mathbf{x} \in \mathbb{R}^n \ \text{and} \ 
\nu(\mathbf{x}) \leq 1 \right\rbrace.}           

\smallskip

We first prove (i). Suppose $\displaystyle m\left(A_{f, \psi, \eta}\right) < \infty.$ In view of \equ{asympproperty}, it follows that for every $M \in \mathbb{Z}_{\geq 1}$ we have $\displaystyle m\left(A_{f, M\psi, \nu}\right) < \infty.$ Theorem \ref{genericcounting}(i) then implies that for every $M \in \mathbb{Z}_{\geq 1}$ we have \[\mu_{X}\left(\left\{\Lambda \in X: {\#}\left(\Lambda_{{\mathcal{P}}} \cap A_{f, M\psi, \nu}\right) = \infty \right\}\right) = 0. \] Hence, the set \[ S_1 :=   \bigcup_{M \in \mathbb{Z}_{\geq 1}}\left\lbrace g \in G: {\#}\left(g\mathcal{P} \cap A_{f, M\psi, \nu}\right) = \infty \right\rbrace \] satisfies $\mu_G({S_1}) = 0$. Now let $g = \langle h, \mathbf{z} \rangle$ be any element of $G$ for which \eq{appr}{f \circ g\text{ is $\left(\psi, \nu, {{\mathcal{P}}}\right)$-approximable}.} Let $\displaystyle D := \max\left\lbrace \|h\|, \norm{h^{-1}}\right\rbrace,$ and let $\displaystyle E := \nu(\mathbf{z}).$ Let $k$ be an element of $\mathbb{Z}_{\geq 1}$ for which $\displaystyle a^{k} > D.$
Since $\psi$ is regular and nonincreasing, it follows from {Lemma \ref{KSLemma3.2}}
that there exists $F \in \mathbb{R}_{>0}$ for which the following is true:~for each $x \in [0, E]$ and each $\displaystyle y \in (E, \infty),$ we have $\psi(y-x) \preccurlyeq F \psi(y).$ Let $N$ be any element of $\mathbb{Z}_{\geq 1}$ with $N > b^{-k}F.$ {A simple argument then yields the inclusion} \eq{nullinclusion}{  {g\left(\left\lbrace \mathbf{x} \in
A_{f \circ g, \psi, \nu}  : \nu(\mathbf{x}) > 2 DE \right\rbrace\right) \subseteq
A_{f, N\psi, \nu}.}} In light of \equ{appr} and \equ{nullinclusion}, it follows that $\displaystyle {\#}\left(g\mathcal{P} \cap A_{f, N\psi, \nu}\right) = \infty.$ We conclude that $g$ belongs to the $\mu_G$-null set ${S_1}$.
This completes the proof of (i).      


\smallskip


{The proof of (ii) proceeds along similar lines}. Suppose $\displaystyle m\left( A_{f, \psi, \eta} \right) = \infty,$  and suppose that we are given $r \in \mathbb{R}_{> 1}$  for which $G$ is of $\displaystyle \left(\mathcal{P}, r\right)$-Rogers type. It then follows from \equ{asympproperty} and Theorem \ref{genericcounting}(ii) that the set  \[ S_2 :=   \bigcup_{M \in \mathbb{Z}_{\geq 1}}\left\lbrace g \in G: {\#}\left(g\mathcal{P} \cap A_{f, \psi/M, \nu}\right) < \infty \right\rbrace \] is $\mu_G$-null.  Now let $g$ be any element of $G$ for which $f \circ g$ is not $\left(\psi, \nu, {{\mathcal{P}}}\right)$-approximable. Arguing as in the proof of (i), we conclude that
{$g\in S_2$, which finishes}
the proof of (ii).  \end{proof}

\begin{rmk}\label{quant} \rm Using \equ{limit} and arguing as in the proof of \cite[Theorem 3.4]{KS}, it is possible to enhance the qualitative conclusion of Theorem \ref{asymptoticmain}(ii) in a quantitative fashion. Since we are primarily interested in qualitative results, we decided to forego quantitative arguments. 
\end{rmk}




\ignore{Let $\varepsilon \in \mathbb{R}_{>0}$ be given. Let $K$ be an arbitrary nonempty compact subset of $G$; we assume without loss of generality that $K = K^{-1}.$ Define $\pi : {\rm ASL}_n(\mathbb{R}) \to \mathrm{SL}_n(\R)$ and $\rho : {\rm ASL}_n(\mathbb{R}) \to \mathbb{R}^n$ by $\displaystyle \pi : \langle h, \mathbf{z} \rangle \mapsto h$ and $\displaystyle \rho : \langle h, \mathbf{z} \rangle \mapsto \mathbf{z}.$ Notice that each of these maps is continuous. We define \eq{constants}{ D_K := \sup \left\lbrace \|h\| : h \in \pi(K)  \right\rbrace < \infty \quad \quad \quad \text{and} \quad \quad \quad E_K := \sup \left\lbrace \nu(\mathbf{z}) : \mathbf{z} \in \rho(K) \right\rbrace < \infty.} Note that $D_K \geq 1.$ 
Set $\displaystyle {k} := \min\left\lbrace j \in \mathbb{Z}_{\geq 0} : a^j \geq D_K \right\rbrace$ and $C_K := b^{-{k}}.$ Note that $C_K \geq 1.$ 
Since $\psi$ is regular and nonincreasing, it follows from \cite[Lemma 3.2]{KS} that there exists $F_K \in \mathbb{R}_{>0}$ for which the following is true:~for each $x \in [0, E_K]$ and each $y \in (E_K, \infty),$ we have $\displaystyle \psi(y-x) \preccurlyeq F_K \psi(y).$ Set $\displaystyle J_K := C_K F_K.$ Since $\psi$ is nonincreasing and its image is a subset of $\left(\mathbb{R}_{>0}\right)^\ell,$ it follows $F_K \geq 1.$ Thus, $J_K \geq 1.$               



\smallskip   


   
Let $g_1 \in K$ be arbitrary. Let $R$ be any real number with $R > 2D_KE _K.$ Arguing as in the proof of \cite[Theorem 3.4]{KS}, we establish that \eq{supinc}{ g_1\left(
\left\lbrace \mathbf{t} \in A_{ f \circ g_1, \psi, \nu } : 2 D_K E_K < \nu(\mathbf{t}) \leq R \right\rbrace \right) \subseteq \left\lbrace \mathbf{t} \in A_{f, J_K \psi, \nu} : E_K < \nu(\mathbf{t}) \leq D_K R + E_K \right\rbrace.} In view of \equ{asympproperty}, we have $\displaystyle m\left(\left\lbrace \mathbf{t} \in A_{f, J_K \psi, \nu} : \nu(\mathbf{t}) > E_K \right\rbrace\right) = \infty.$ Using \equ{supinc} and then applying \cite[Theorem 2.9(ii)]{KS}, it follows that for $\mu_G$-almost every $g \in K$ and any $\varepsilon \in \mathbb{R}_{>0}$ there exists some $\displaystyle T_g \in \mathbb{R}_{>0}$ such that for every $T \in \left(T_g, \infty\right),$ we have \begin{align*}
&\frac{{\#}\left\lbrace \mathbf{v} \in {{\mathcal{P}}} : (f \circ g)(\mathbf{v}) \preccurlyeq \psi\big( \nu(\mathbf{v}) \big) \ \text{and} \ 2 D_K E_K < \nu(\mathbf{v}) \leq T \right\rbrace}{m\left(\left\lbrace \mathbf{t} \in \mathbb{R}^n : f(\mathbf{t}) \preccurlyeq J_K\psi\big(\nu(\mathbf{t})\big) \ \text{and} \ {E_K} < \nu(\mathbf{t}) \leq D_K T + E_K \right\rbrace\right)} \\
&\leq \frac{{\#}\left\lbrace \mathbf{w} \in  g {{\mathcal{P}}} : f(\mathbf{w}) \preccurlyeq J_K\psi\big(\nu(\mathbf{w})\big) \ \text{and} \ {E_K} < \nu(\mathbf{w}) \leq D_K T + E_K \right\rbrace}{m\left(\left\lbrace \mathbf{t} \in \mathbb{R}^n : f(\mathbf{t}) \preccurlyeq J_K\psi(\nu\big(\mathbf{t})\big) \ \text{and} \ {E_K} < \nu(\mathbf{t}) \leq D_K T + E_K \right\rbrace\right)} \\
&< {{c}_{{\mathcal{P}}}} + \varepsilon.
\end{align*}
It follows that \eqref{limsup} holds for $\mu_G$-almost every $g \in K$. A similar argument allows us to conclude that \eqref{liminf} holds for  $\mu_G$-almost every $g \in K$. It is clear from \equ{constants} that $E_K = 0$ if and only if $K \subseteq \mathrm{SL}_n(\R)$, which proves the penultimate statement of (ii). The final statement of (ii) follows from the $\sigma$-compactness of $G.$   




Now let $g_2 \in K$ be arbitrary. Let $R'$ be any real number with $R' > 2E_K.$ We then have \eq{infinc}{ g_2^{-1}\left(
\left\lbrace \mathbf{t} \in  A_{f, J_K^{-1}\psi, \nu} : 2 E_K <\nu(\mathbf{t}) \leq R' \right\rbrace\right)
 \subseteq \left\lbrace \mathbf{t} \in A_{f \circ g_2, \psi, \nu} : {E_K D_K^{-1}} < \nu(\mathbf{t}) \leq D_K\left(E_K + R'\right) \right\rbrace.} The proof of \equ{infinc} is very similar to that of \equ{supinc}.\footnote{The proof of \cite[(3.8)]{KS}, which corresponds to \equ{infinc} above, was omitted from the proof of \cite[Theorem 3.4]{KS} because of its similarity to the detailed proof of \cite[(3.7)]{KS}, which corresponds to \equ{supinc} above. A detailed proof of \equ{infinc} can be found in the dissertation of the co-author Skenderi:~see the proof of \cite[Theorem 6.4]{SkeDis}.} In view of \equ{asympproperty}, we have $$ m\left(\left\lbrace \mathbf{t} \in  A_{f, J_K^{-1}\psi, \nu} : \nu(\mathbf{t}) > 2 E_K \right\rbrace\right) = \infty.$$ Using \equ{infinc} and then applying \cite[Theorem 2.9 (ii)]{KS}, it follows that for $\mu_G$-almost every $g \in K$ and any $\varepsilon \in \mathbb{R}_{>0}$ there exists some $\displaystyle T_{g}' \in \mathbb{R}_{>0}$ such that for every $T \in \left( T_{g}', \infty\right),$ we have  \begin{align*}
&\frac{{\#}\left\lbrace \mathbf{v} \in {{\mathcal{P}}} : (f \circ g)(\mathbf{v}) \preccurlyeq \psi\big( \nu(\mathbf{v})\big) \ \text{and} \ {E_K D_K^{-1}} < \nu(\mathbf{v}) \leq D_K\left(E_K + T\right) \right\rbrace}{m\left(\left\lbrace \mathbf{t} \in \mathbb{R}^n : f(\mathbf{t}) \preccurlyeq J_K^{-1}\psi\big(\nu(\mathbf{t})\big) \ \text{and} \ 2E_K < \nu(\mathbf{t}) \leq T \right\rbrace\right)} \\
&\geq \frac{{\#}\left\lbrace \mathbf{w} \in g{{\mathcal{P}}} : f(\mathbf{w}) \preccurlyeq J_K^{-1}\psi\big(\nu(\mathbf{w})\big) \ \text{and} \ 2E_K < \nu(\mathbf{w}) \leq T \right\rbrace}{m\left(\left\lbrace \mathbf{t} \in \mathbb{R}^n : f(\mathbf{t}) \preccurlyeq J_K^{-1}\psi\big(\nu(\mathbf{t})\big) \ \text{and} \ 2E_K < \nu(\mathbf{t}) \leq T \right\rbrace\right)} \\
&> {{c}_{{\mathcal{P}}}} - \varepsilon.
\end{align*}
It follows that \eqref{liminf} holds for  $\mu_G$-almost every $g \in K$. It is clear from \equ{constants} that $E_K = 0$ if and only if $K \subseteq \left( \mathrm{SL}_n(\R) \times \left\lbrace \mathbf{0}_{\mathbb{R}^n} \right\rbrace\right)$, which proves the penultimate statement of (ii). The final statement of (ii) now follows simply because $G$ is $\sigma$-compact.   }
To state the next theorem, we need the following definition.       

\begin{defn}\label{subsequences}
\begin{itemize} \item[]
\item Let $\displaystyle t_\bullet = \left(t_k\right)_{k \in \mathbb{Z}_{\geq 1}}$ be any strictly increasing sequence of elements of $\mathbb{R}_{>0}$ with $\displaystyle \lim_{k \to \infty} t_k = \infty.$ We then say that $f$ is $\displaystyle t_\bullet$-\textsl{uniformly} $\left( \psi, \nu, \mathcal{P} \right)$-approximable if {$B_{f, \psi(t_k), \nu, t_k} \cap \mathcal{P}  \neq \varnothing$ for each sufficiently large $k \in \mathbb{Z}_{\geq 1}$}.   
\item Let $\displaystyle t_\bullet = \left(t_k\right)_{k \in \mathbb{Z}_{\geq 1}}$ be any strictly increasing sequence of elements of $\mathbb{R}_{>0}$ with $\displaystyle \lim_{k \to \infty} t_k = \infty.$ We say that $t_\bullet$ is \textsl{quasi-geometric} if, in addition to the preceding, the set $\displaystyle \left\lbrace t_{k+1}/t_k : k \in \mathbb{Z}_{\geq 1} \right\rbrace$ is bounded. 
\end{itemize}   
\end{defn}


\begin{thm}\label{mainuniformresult}
Let $f : \mathbb{R}^n \to \mathbb{R}^{\ell}$ and $\psi : \mathbb{R}_{\geq 0} \to \left(\mathbb{R}_{>0}\right)^\ell$ each be Borel measurable. Suppose that the pair $(f, \psi)$ is uniformly acceptable. Let $u_\bullet = (u_k)_{k \in \mathbb{Z}_{\geq 1}}$ be any strictly increasing sequence of elements of $\mathbb{R}_{>1}$ with $\displaystyle \lim_{k \to \infty} u_k = \infty.$ Let $G$ be a closed subgroup of ${ \rm ASL}_n(\mathbb{R})$, let $\Gamma$ be as in $\equ{gamma},$ and let ${{\mathcal{P}}}$ be a $\Gamma$-invariant subset of $\mathbb{Z}^n.$ Suppose $G$ is of $\mathcal{P}$-Siegel type, and suppose that we are given $r \in \mathbb{R}_{>1}$ for which $G$ is of $\left(\mathcal{P}, r\right)$-Rogers type. Suppose that there exists some norm $\eta$ on $\mathbb{R}^n$ for which \eq{conv}{\inf_{N \in \mathbb{Z}_{\geq 1}}\sum_{k=N}^{\infty}  m\left(B_{f, \psi\left(u_k\right), \eta, u_k}\right)^{1-r} < \infty.} Let $\nu$ be an arbitrary norm on $\mathbb{R}^n$. The following then hold. 
\vspace{0.055in}
\begin{itemize}
\item[\rm (i)] For almost every $g \in G$ the function $f \circ g$ is $u_\bullet$-uniformly $\displaystyle \left(\psi, \nu, \mathcal{P}\right)$-approximable. 
\vspace{0.055in}
\item[\rm (ii)] Suppose further that $\psi$ is regular and nonincreasing and that the sequence $u_\bullet$ is quasi-geometric. Then for almost every $g \in G$ the function $f \circ g$ is uniformly $(\psi, \nu, {{\mathcal{P}}})$-approximable. \end{itemize} \end{thm}

\begin{proof} 
We argue as in the proof of \cite[Theorem 3.8]{KS}, appealing to the uniform acceptability of the pair $(f,\psi)$ (instead of appealing to Lemma \ref{3.1and3.5}).     

\vspace{0.050in}   
\begin{itemize}
\item[\rm (i)] As in the proof of Theorem \ref{asymptoticmain}, we denote elements of $\mathrm{ASL}_n(\mathbb{R})$ by $\langle h,\mathbf{z} \rangle.$ For any $h \in \mathrm{SL}_n(\R)$, let $\|h\|$ denote the operator norm of $h$
given by {\equ{norm}}.
Define $\pi : {\rm ASL}_n(\mathbb{R}) \to \mathrm{SL}_n(\R)$ and $\rho : {\rm ASL}_n(\mathbb{R}) \to \mathbb{R}^n$ by $\displaystyle \pi : \langle h, \mathbf{z} \rangle \mapsto h$ and $\displaystyle \rho : \langle h, \mathbf{z} \rangle \mapsto \mathbf{z}.$ Notice that each of these maps is continuous. Let $K$ be a  nonempty compact subset of $G$ with $K = K^{-1}$, and define \[ { D_K := \sup \left\lbrace \|h\| : h \in \pi(K)  \right\rbrace < \infty \quad \quad \quad \text{and} \quad \quad \quad E_K := \sup \left\lbrace \nu(\mathbf{z}) : \mathbf{z} \in \rho(K) \right\rbrace < \infty.} \] Fix $L \in \mathbb{Z}_{\geq 1}$ such that for each $k \in \mathbb{Z}_{\geq L}$ we have $u_k > 2 E_K.$ 
It follows from \equ{unifproperty} and \equ{conv} that $\displaystyle \inf_{N \in \mathbb{Z}_{\geq 1}}\sum_{k=N}^{\infty}  m\left(B_{f, \psi\left(u_k\right), \nu , {u_k}/{2D_K} }\right)^{1-r} < \infty.$ We then apply Theorem \ref{genericcounting}(iii) to obtain the following:~for almost every $g \in G$ there exists $M_{g} \in \mathbb{Z}_{\ge L}$ such that for each $k \in \mathbb{Z}$ with $k \geq M_g$ there exists some $\mathbf{v}_{g, k} \in \mathcal{P}$ with \begin{equation}\label{somelabel} \nu\left(g \mathbf{v}_{g, k}\right) \leq \frac{u_k}{2D_K} \quad \quad  \quad \text{and} \quad \quad \quad  \left|f\left(g\mathbf{v}_{g, k}\right)\right| \preccurlyeq \psi\left(u_k\right).   
\end{equation} {For any such $\mu_G$-generic $g$ that belongs to $K$ and any integer $k \geq M_g,$ we have $\nu\left(\mathbf{v}_{g, k}\right) \leq u_k$; this may be proved by appealing to the first inequality in \eqref{somelabel} and arguing as in the proof of \cite[Theorem 3.8(i)]{KS}. We thus conclude that for $\mu_G$-almost every $g \in K$ the function $f \circ g$ is $u_\bullet$-uniformly $(\psi, \nu, \mathcal{P})$-approximable. Since $G$ is $\sigma$-compact, the desired result follows. }


\smallskip


\item[\rm (ii)] Let $a = a(\psi) \in \mathbb{R}_{>1}$ and $b = b(\psi) \in \mathbb{R}_{>0}$ be as in Definition \ref{basicapdefns}. Fix $j \in \mathbb{Z}_{\geq 1}$ for which $\displaystyle \sup\left\lbrace {u_{k+1} /u_k} : k \in \mathbb{Z}_{\geq 1} \right\rbrace < a^j.$ (This is possible because $u_\bullet$ is quasi-geometric.) Appealing once again to \equ{unifproperty} and \equ{conv}, we infer $\displaystyle \inf_{N \in \mathbb{Z}_{\geq 1}} \sum_{k=N}^{\infty}  (m\left(B_{f, b^j\psi\left(u_k\right), \eta, u_k}\right) ^{1-r} < \infty.$ Statement (i) of this theorem implies that for almost every $g \in G$ the function $f \circ g$ is $u_\bullet$-uniformly $\left(b^j\psi, \nu, \mathcal{P}\right)$-approximable. Now let $\displaystyle h : \mathbb{R}^n \to \mathbb{R}^\ell$ be any function that is $u_\bullet$-uniformly $\left(b^j\psi, \nu, \mathcal{P}\right)$-approximable. Fix ${M} \in \mathbb{Z}_{\geq 1}$ such that for each $k \in \mathbb{Z}_{\geq {M} }$ the set $\displaystyle B_{h, b^j\psi(u_k), \nu, u_k} \cap \mathcal{P}$ is nonempty. Let $\displaystyle T \in \left( u_{{M} +2}, + \infty\right)$ be arbitrary. Then there exists $i \in \mathbb{Z}_{\geq {M} +2}$ for which $\displaystyle u_i \leq T \leq u_{i+1}.$ Note that there exists $\mathbf{v} \in \mathcal{P}$ with $\nu(\mathbf{v}) \leq u_i$ and $\displaystyle |h(\mathbf{v})| \preccurlyeq b^j \psi(u_i).$ We then have $\nu(\mathbf{v}) \leq u_i \leq T$ and  \begin{equation*} |h(\mathbf{v})| \preccurlyeq b^j \psi(u_i) \preccurlyeq b^j b^{-j} \psi\left(a^j u_i \right) = \psi\left(a^j u_i \right) \preccurlyeq \psi\left(u_{i+1}\right) \preccurlyeq \psi(T). \end{equation*}  This completes the proof. \end{itemize}  \end{proof} 

\begin{rmk} \rm 
The infimum in \equ{conv} is included because $\sum_{k=1}^{\infty}  m\left(B_{f, \psi\left(u_k\right), \eta, u_k}\right)^{1-r}$ may diverge for the trivial reason that there exist finitely many $k \in \mathbb{Z}_{\geq 1}$ for which $m\left(B_{f, \psi\left(u_k\right), \eta, u_k}\right) = 0.$         
\end{rmk}    

\smallskip

\section{Applications of general results}\label{acc}

We begin with the following elementary observation.

\begin{lem}\label{local}
Let $\left(M, g_M\right)$ be an oriented $\mathscr{C}^1$ Riemannian manifold that is Hausdorff, second-countable, and without boundary. Let $\sigma$ denote the Borel measure on $M$ induced by the natural Riemannian volume form on $M$. Let $h : M \to \mathbb{R}^{\ell}$ be a $\mathscr{C}^1$ map, and suppose that $h^{-1}\left(\mathbf{0}_{\mathbb{R}^\ell}\right) \neq \varnothing.$ Let $z \in h^{-1}\left(\mathbf{0}_{\mathbb{R}^\ell}\right)$, and suppose that $z$ is a regular point of $h.$ Then there exist $C = C_z \in \mathbb{R}_{>1}$, an open subset $V = V_z$ of $\mathbb{R}^\ell$ with $\mathbf{0}_{\mathbb{R}^\ell} \in V$, and an open subset $W = W_z$ of $M$ with $z \in W$ such that for any Borel subset $E$ of $\mathbb{R}^\ell$ with $E \subseteq V,$ we have \begin{equation*}
C^{-1} \, m(E) \leq \sigma\big(W\cap h^{-1}(E)\big) \leq C \, m(E). 
\end{equation*}
In particular, $m|_{V}$ and the restriction to $V$ of the pushforward of $\sigma|_{W}$ by $h|_{W}$ are equivalent Borel measures.  \end{lem}  


\begin{proof} For the sake of clarity, we note that $m$ denotes Lebesgue measure on $\mathbb{R}^\ell$. Set $k := \dim M.$ Note that $k \geq \ell.$ By the Constant Rank Theorem \cite[Theorem 7.1]{Boo} 
there exist $\varepsilon \in \mathbb{R}_{>0}$ and maps $\phi:(-\varepsilon,\varepsilon)^k\to M$ and $\Phi:\mathbb{R}^\ell \to \mathbb{R}^\ell$ such that:

\begin{itemize} 
\item the set $\phi\left((-\varepsilon,\varepsilon)^k\right)$ is an open subset of $M$, and $\phi$ is a $\mathscr{C}^1$ diffeomorphism onto $\phi\left((-\varepsilon,\varepsilon)^k\right)$;
\item the set $\Phi\left(\mathbb{R}^\ell\right)$ is an open subset of $\mathbb{R}^\ell$, and $\Phi$ is a $\mathscr{C}^1$ diffeomorphism onto $\Phi\left(\mathbb{R}^\ell\right)$;
\item $\phi\left(\mathbf{0}_{\mathbb{R}^k}\right) = z$; and
\item $\Phi\circ h\circ\phi = {\pi_\ell}_{|(-\varepsilon,\varepsilon)^k}$, where $\pi_\ell : \mathbb{R}^k\to\mathbb{R}^\ell$ is given by $(x_1,\dots,x_k) \mapsto (x_1,\dots,x_\ell)$.
\end{itemize}
\vspace{0.02in} 
Set $\displaystyle W = W_z := \phi\left( \left( -\frac{\varepsilon}{2}, \frac{\varepsilon}{2} \right)^k \right) \subseteq M.$ Then $W$ is an open subset of $M$ for which \[  z \in W \subset \overline{W} = \phi\left( \left[ -\frac{\varepsilon}{2}, \frac{\varepsilon}{2} \right]^k \right) \subset \phi\left((-\varepsilon,\varepsilon)^k\right) \subseteq M. \] Note that $\Phi\left(\mathbf{0}_{\mathbb{R}^\ell}\right) = \mathbf{0}_{\mathbb{R}^\ell}.$ Set $\displaystyle V = V_z := \Phi^{-1}\left( \left( -\frac{\varepsilon}{2}, \frac{\varepsilon}{2} \right)^\ell \right) \subseteq \mathbb{R}^\ell.$ Then $V$ is an open subset of $\mathbb{R}^\ell$ for which \[ \mathbf{0}_{\mathbb{R}^\ell} \in V \subset \overline{V} = \Phi^{-1}\left( \left[ -\frac{\varepsilon}{2}, \frac{\varepsilon}{2} \right]^\ell \right) \subset \Phi^{-1}\left( \left( -\varepsilon, \varepsilon \right)^\ell \right) \subseteq \mathbb{R}^\ell. \]     

Let $E$ be an arbitrary Borel subset of $\mathbb{R}^\ell$ with $E \subseteq V.$ Then \[ W\cap h^{-1}(E) = \phi\left( \left( -\frac{\varepsilon}{2}, \frac{\varepsilon}{2} \right)^k  \, \cap \,  \pi_\ell^{-1}\big(\Phi(E)\big)\right). \] Since each of $\overline{V}$ and $\overline{W}$ is compact, each of $h$ and $\pi_\ell$ is of class $\mathscr{C}^1$, and each of $\phi$ and $\Phi$ is a $\mathscr{C}^1$ diffeomorphism from its domain onto its image, it follows that \[ \sigma\big(W\cap h^{-1}(E)\big) \asymp_{{g_M, \, z}} \, m\left(  \left( -\frac{\varepsilon}{2}, \frac{\varepsilon}{2} \right)^k  \, \cap \,  \pi_\ell^{-1}\big(\Phi(E)\big)\right) = \varepsilon ^{k-\ell} \, m\big(\Phi(E)\big) \asymp_{z} \, m(E). \] \qedhere   \end{proof}

Let us now use the preceding lemma to derive a global statement. 

\begin{thm}\label{submersionvol}
Let $\left(M, g_M\right)$ be an oriented $\mathscr{C}^1$ Riemannian manifold that is compact, Hausdorff, second-countable, and without boundary. Let $\sigma$ denote the Borel measure on $M$ induced by the natural Riemannian volume form on $M$. Let $h : M \to \mathbb{R}^{\ell}$ be a $\mathscr{C}^1$ map; suppose that $h^{-1}\left(\mathbf{0}_{\mathbb{R}^\ell}\right) \neq \varnothing$ and that every element of $h^{-1}\left(\mathbf{0}_{\mathbb{R}^\ell}\right)$ is a regular point of $h.$ Then there exists an open subset $V$ of $\mathbb{R}^\ell$ with $\mathbf{0}_{\mathbb{R}^\ell} \in V$ such that for any Borel subset $E$ of $\mathbb{R}^\ell$ with $E \subseteq V,$ we have  \[ {\sigma\big(  h^{-1}(E)\big)} \asymp_{M, {g_M}, h} \, {m(E)}. \]  \end{thm}  \begin{proof} 
Set $Z := h^{-1}\left(\mathbf{0}_{\mathbb{R}^\ell}\right) \neq \varnothing.$ For every $z \in Z,$ let $C_z \in \mathbb{R}_{>1},$ $V_z \subseteq \mathbb{R}^\ell$ with $\mathbf{0}_{\mathbb{R}^\ell} \in V_z,$ and $W_z \subseteq M$ with $z \in W_z$ be as in Lemma \ref{local}. Since $Z$ is compact, there exist finitely many $z_1, \dots , z_N \in Z$ such that $\displaystyle Z \subseteq W := \bigcup_{i=1}^N W_{z_i}. $ Set $\displaystyle U := \bigcap_{i=1}^N V_{z_i}$ and $\displaystyle C := \sum_{i=1}^N C_{z_i}.$ Let $V$ be an open subset of $\mathbb{R}^\ell$ such that $\displaystyle \mathbf{0}_{\mathbb{R}^\ell} \in V \subseteq U$ and $h^{-1}(V) \subseteq W.$ (We defer the proof of the existence of $V$ until the end of this theorem's proof.) If $E$ is any Borel subset of $\mathbb{R}^\ell$ with $E \subseteq V,$ then \[ C^{-1} \, m(E) \le \min \big\{\sigma\big(W_{z_i}\cap h^{-1}(E)\big): 1\le i \le N\big\} \le \sigma\big(  h^{-1}(E)\big)\le  \sum_{i=1}^N \sigma\big(W_{z_i}\cap h^{-1}(E)\big)\le C \, m(E). \] 

We now prove the existence of such a set $V$; suppose by way of contradiction that such a set did not exist. This would imply that for every open subset $U'$ of $\mathbb{R}^\ell$ with $\displaystyle \mathbf{0}_{\mathbb{R}^\ell} \in U' \subseteq U$ there exists $\mathbf{y} \in (M \ssm W)$ for which $h(\mathbf{y}) \in U'$. Let $\|\cdot\|$ denote the Euclidean norm on $\mathbb{R}^\ell.$ Then for each $r \in \mathbb{Z}_{\geq 1}$ there exists some $\mathbf{x}_r \in (M \ssm W)$ for which $h(\mathbf{x}_r) \in U$ and $\| h(\mathbf{x}_r) \| < r^{-1}$. Since $M$ is compact, the sequence $\displaystyle \left( \mathbf{x}_r \right)_{r \in \mathbb{Z}_{\geq 1}}$ has a convergent subsequence whose limit we denote by $\mathbf{x} \in M.$ Since $W$ is an open subset of $M$, we have $\mathbf{x} \in (M \ssm W)$; this implies $\mathbf{x} \notin Z$, so that $h(\mathbf{x}) \neq \mathbf{0}_{\mathbb{R}^\ell}.$ On the other hand, the sequence $\displaystyle \left( h(\mathbf{x}_r) \right)_{r \in \mathbb{Z}_{\geq 1}}$ clearly converges to $\mathbf{0}_{\mathbb{R}^\ell}$. The continuity of $h$ then implies $h(\mathbf{x}) =  \mathbf{0}_{\mathbb{R}^\ell}.$ This is a contradiction. \qedhere  \end{proof}


\begin{standing} { Let us state here the conventions that will be in force throughout the remainder of this paper. }
\begin{itemize}
\item We shall let $G$ denote a closed subgroup of $\mathrm{ASL}_n(\mathbb{R})$ and let $\mathcal{P}$ denote a $\Gamma$-invariant subset of $\mathbb{Z}^n,$ where $\Gamma$ is as \equ{gamma}; 

\item we shall assume that $G$ is of $\mathcal{P}$-Siegel type and that we are given $r \in \mathbb{R}_{>1}$ for which $G$ is of $\displaystyle \left(\mathcal{P}, {r}\right)$-Rogers type; 
\item we shall let $\nu$ denote an arbitrary norm on $\mathbb{R}^n.$ 
\end{itemize}  
\end{standing} 



\smallskip



We now state and prove Theorem \ref{submerapprox}, of which Theorem \ref{simplesubmerapprox} is an immediate consequence. \begin{thm}\label{submerapprox}
Let $\displaystyle \psi = (\psi_1, \dots , \psi_{\ell}) : \mathbb{R}_{\geq 0} \to \left(\mathbb{R}_{>0}\right)^{\ell}$ be regular and nonincreasing. Let $\displaystyle f = (f_1, \dots , f_{\ell}) 
: \mathbb{R}^n \to \mathbb{R}^{\ell}$ be homogeneous of degree $\mathbf{d} = \mathbf{d}(f) =  (d_1, \dots, d_\ell)\in \left(\mathbb{R}_{>0}\right)^\ell.$ Suppose further that $f$ is continuously differentiable on $\mathbb{R}^n_{\neq 0}$, that $\mathcal{Z}(f) \neq \varnothing$, and that each element of $\mathcal{Z}(f)$ is a regular point of $f.$ 
Let ${\pmb \xi} \in \mathbb{R}^\ell.$ Set $d := \sum_{j=1}^\ell d_j.$ Then the following hold. 
\begin{itemize}
\item[ \rm (i)] If $\displaystyle \int_1^{\infty} t^{n-(d+1)} \left(\prod_{j=1}^\ell \psi_j(t) \right) \mathrm{d} t$ is finite $($respectively, infinite$)$,
then $\left({_{\pmb \xi}} f\right) \circ g$ is $\left(\psi, \nu, \mathcal{P}\right)$-approximable for Haar-almost no $($respectively, almost every$)$ $g \in G$. 
\item[\rm (ii)] Suppose that $d < n$ and that the infinite series $\displaystyle \sum_{k=1}^{\infty}\left[ 2^{k(n-d)} \prod_{j=1}^\ell \psi_j\left(2^k\right) \right]^{1-r}$ converges. Then $\left({_{\pmb \xi}} f\right) \circ g$ is {\sl uniformly} $\left(\psi, \nu, \mathcal{P}\right)$-approximable for Haar-almost every $g \in G$.
\end{itemize}
\end{thm}

\begin{proof} We begin by attending to some preliminary matters. 


Let $\sigma_n$ denote the unique $\mathrm{SO}(n)$-invariant Radon probability measure on $\mathbb{S}^{n-1} \subset \mathbb{R}^n.$ Let $\|\cdot\|$ denote the Euclidean norm on $\mathbb{R}^n.$ Define $h : \mathbb{S}^{n-1} \to \mathbb{R}^\ell$ to be the restriction of $f$ to $\mathbb{S}^{n-1}.$ 
Note that $h^{-1}\left(\mathbf{0}_{\mathbb{R}^\ell}\right) = \mathcal{Z}(f) \cap \mathbb{S}^{n-1} \neq \varnothing.$ Now let $\mathbf{x} \in h^{-1}\left(\mathbf{0}_{\mathbb{R}^\ell}\right).$ 
For each $j \in \{1, \dots , \ell \},$ the homogeneity of $f_j$ implies that $\nabla f_j (\mathbf{x})$ is tangent to $\mathbb{S}^{n-1}$. It follows that $\mathbf{x}$ is a regular point of $f : \mathbb{R}^n \to \mathbb{R}^\ell$ if and only if $\mathbf{x}$ is a regular point of $h : \mathbb{S}^{n-1} \to \mathbb{R}^\ell.$ We thereby conclude that every element of $h^{-1}\left(\mathbf{0}_{\mathbb{R}^\ell}\right)$ is a regular point of $h$. Theorem \ref{submersionvol} may thus be applied to $h : \mathbb{S}^{n-1} \to \mathbb{R}^\ell.$ Let $V \subseteq \mathbb{R}^\ell$ be an open neighborhood of $\mathbf{0}_{\mathbb{R}^\ell}$ as in the conclusion of Theorem \ref{submersionvol}. 

Let us now introduce a pair of mutually inverse bijections that we shall use in this proof:
\begin{align}
&\mathbb{R}_{>0} \times \mathbb{S}^{n-1} \to \mathbb{R}^n_{\neq 0} \quad \quad \text{given by} \quad \quad (t, \mathbf{u}) \mapsto t\mathbf{u} \label{bijectone} \\
&\text{and} \nonumber \\ 
&\mathbb{R}^n_{\neq 0} \to \mathbb{R}_{>0} \times \mathbb{S}^{n-1}\quad \quad \text{given by} \quad \quad \mathbf{x} \mapsto \left( \|\mathbf{x}\|, \frac{\mathbf{x}}{\|\mathbf{x}\|} \right). \label{bijecttwo}
\end{align}


Finally, for each $t \in \mathbb{R}_{>0},$ define \[ g_t := \mathrm{diag}\left( t^{-d_1}, \dots , t^{-d_\ell}\right) \in \mathrm{GL}_\ell(\mathbb{R}). \]   


\begin{itemize}
\item[(i)] Let $s \in \mathbb{R}_{>0}.$ We shall show that the pair $\left({_{\pmb \xi} f}, \psi\right)$ is asymptotically acceptable by first showing that 
\eq{checkasymp}{
m\left(A_{ _{\pmb \xi} f, s\psi, \|\cdot\|}\right) < \infty \quad \text{if and only if} \quad \int_1^{\infty} t^{n-(d+1)} \left(\prod_{j=1}^\ell \psi_j(t) \right) \mathrm{d} t < \infty.  
} Using the bijections in \eqref{bijectone} and \eqref{bijecttwo} and the homogeneity of $f$, it follows that for each $t \in \mathbb{R}_{>0}$ and each $\mathbf{u} \in \mathbb{S}^{n-1}$ we have \begin{equation}\label{sphereiff} t\mathbf{u} \in A_{ _{\pmb \xi} f, s\psi, \|\cdot\|}\quad \text{if and only if}\quad g_t \big( {\pmb \xi} - s\psi({t}) \big)  \preccurlyeq h(\mathbf{u}) \preccurlyeq g_t \big( {\pmb \xi} + s \psi({t}) \big). 
\end{equation}  Using the boundedness of $\psi$, we now fix $M \in \mathbb{R}_{> 2}$ such that for each $t \in \mathbb{R}_{\geq M}$ we have \begin{equation*}
\left\lbrace \mathbf{w} \in \mathbb{R}^\ell :  g_t \big( {\pmb \xi} - s\psi({t}) \big)  \preccurlyeq \mathbf{w} \preccurlyeq g_t \big( {\pmb \xi} + s \psi({t}) \big)   \right\rbrace \subseteq V . \end{equation*} Theorem \ref{submersionvol} then implies that for each $t \in \mathbb{R}_{\geq M}$ we have  \begin{equation}\label{followsfrom}
\sigma_n\left( \left\lbrace \mathbf{u} \in \mathbb{S}^{n-1} : g_t \big( {\pmb \xi} - s\psi({t}) \big)  \preccurlyeq h(\mathbf{u}) \preccurlyeq g_t \big( {\pmb \xi} + s \psi({t}) \big)  \right\rbrace \right) \asymp_{n, \ell, f} \ t^{-d} \, \left(\left(2 s\right)^\ell \, \prod_{j=1}^\ell \psi_j\left(t\right)\right).
\end{equation} It then follows from \eqref{sphereiff} and \eqref{followsfrom} that \begin{align*} &m\left(\{ \mathbf{x} \in A_{ _{\pmb \xi} f, s\psi, \|\cdot\|} :   \|\mathbf{x}\| \geq M \}\right) \\
&\asymp_n \ \int_{M}^{{\infty}} \,  t^{n-1} \  \sigma_n\left( \left\lbrace \mathbf{u} \in \mathbb{S}^{n-1} : g_t \big( {\pmb \xi} - s\psi({t}) \big)  \preccurlyeq h(\mathbf{u}) \preccurlyeq g_t \big( {\pmb \xi} + s \psi({t}) \big)  \right\rbrace \right) \mathrm{d} t \\
&\asymp_{n, \ell, f} \ \left(2   s\right)^\ell \int_{M}^{{\infty}} \,  t^{n-(d+1)} \, \left( \prod_{j=1}^\ell \psi_j\left(t\right) \right) \mathrm{d} t.  \end{align*} 
Since $\psi$ is bounded, this proves \equ{checkasymp}. Lemma \ref{scalingonly}(i) then implies that the pair $\left({_{\pmb \xi} f}, \psi\right)$ is asymptotically acceptable. The desired result now follows from Theorem \ref{asymptoticmain}. 

\ignore{If $\displaystyle m\left(A_{ _{\pmb \xi} f, s\psi, \|\cdot\|}\right) < \infty,$ then $\displaystyle \sum_{k=1}^{\infty} m\left(\{ \mathbf{x} \in A_{ _{\pmb \xi} f, s\psi({k+1}), \|\cdot\|} : k \leq \|\mathbf{x}\| \leq {k+1} \}\right) < \infty$ because the function $s\psi$ is nonincreasing.  Now, using  the boundedness and the regularity of $\psi$,  take $k$ large enough so that for $t\ge k$ we have \begin{equation*}
\left\lbrace \mathbf{w} \in \mathbb{R}^\ell :  g_t \big( {\pmb \xi} - s\psi({k+1}) \big)  \preccurlyeq \mathbf{w} \preccurlyeq g_t \big( {\pmb \xi} + s \psi({k+1}) \big)   \right\rbrace \subseteq V  \end{equation*} 
and 
$$b\psi({t}) \preccurlyeq \psi(t+1),$$
where $b = b(\psi) \in \mathbb{R}_{>0}$ is as in Definition \ref{basicapdefns}. Then 

\begin{align*}&\geq \int_{k}^{{k+1}} \,  t^{n-(d+1)} \, \left(\left(2 b^i s\right)^\ell \, \prod_{j=1}^\ell \psi_j\big(a^{-i}\left(k+1\right)\big) \right) \mathrm{d} t \\
&\geq \left(2 b^i s\right)^\ell \int_{k}^{{k+1}} \,  t^{n-(d+1)} \, \left( \prod_{j=1}^\ell \psi_j\left(t\right) \right) \mathrm{d} t,  \end{align*} where the last inequality holds because $a^{-i} < 1/3.$ We thus conclude that if $\displaystyle m\left(A_{ _{\pmb \xi} f, s\psi, \|\cdot\|}\right) < \infty,$ then $\displaystyle \int_1^{\infty} t^{n-(d+1)} \left(\prod_{j=1}^\ell \psi_j(t) \right) \mathrm{d} t < \infty.$    

\smallskip     

We now prove the other direction of the claimed equivalence. If $\displaystyle m\left(A_{ _{\pmb \xi} f, s\psi, \|\cdot\|}\right) = \infty,$ then $\displaystyle \sum_{k=1}^{\infty} m\left(\{ \mathbf{x} \in A_{ _{\pmb \xi} f, s\psi({k}), \|\cdot\|} : k \leq \|\mathbf{x}\| \leq {k+1} \}\right) = \infty.$ As before, it follows that for each sufficiently large $k \in \mathbb{Z}_{\geq 1}$ we have  \begin{align*}
m\left(\{ \mathbf{x} \in A_{ _{\pmb \xi} f, s\psi({k}), \|\cdot\|} : k \leq \|\mathbf{x}\| \leq {k+1} \}\right) &\asymp_{n, \ell, f} \  \int_{k}^{{k+1}} \,  t^{n-1} \, t^{-d} \, \left(\left(2 s\right)^\ell \, \prod_{j=1}^\ell \psi_j\left(k\right) \right) \mathrm{d} t \\
&\leq \int_{k}^{{k+1}} \,  t^{n-(d+1)} \, \left(\left(2 b^{-i} s\right)^\ell \, \prod_{j=1}^\ell \psi_j\left(a^{i}k\right) \right) \mathrm{d} t \\
&\leq \left(2 b^{-i} s\right)^\ell   \int_{k}^{{k+1}} \,  t^{n-(d+1)} \, \left( \prod_{j=1}^\ell \psi_j\left(t\right) \right) \mathrm{d} t.  \end{align*}  We conclude that if $\displaystyle m\left(A_{ _{\pmb \xi} f, s\psi, \|\cdot\|}\right) = \infty,$ then $\displaystyle \int_1^{\infty} t^{n-(d+1)} \left(\prod_{j=1}^\ell \psi_j(t) \right) \mathrm{d} t = \infty.$
   

We have thus shown that \equ{checkasymp} holds for an arbitrary $s \in \mathbb{R}_{>0}$. Lemma \ref{scalingonly}(i) then implies that the pair $\left({_{\pmb \xi} f}, \psi\right)$ is asymptotically acceptable. The desired result now follows from Theorem \ref{asymptoticmain}.  }     

\vspace{0.1in}

\item[(ii)] Let us prove that the pair $\left({_{\pmb \xi} f}, \psi\right)$ is uniformly acceptable. Let $s_1, s_2 \in \mathbb{R}_{>0}$ be given. Arguing as in part (i), we infer that there exists $M \in \mathbb{R}_{>2}$ such that for each $T \in \mathbb{R}$ with $T> M/s_2$, we have      
\begin{align*}
&m\left( \left\lbrace \mathbf{x} \in B_{{_{\pmb \xi} f}, s_1\psi(T), \|\cdot\|, {s_2T}} :  \|\mathbf{x}\| \geq M \right\rbrace \right) \\  
&\asymp_n \int_{M}^{s_2T} \,  t^{n-1} \  \sigma_n\left( \left\lbrace \mathbf{u} \in \mathbb{S}^{n-1} :  g_t\left( {\pmb \xi} - s_1\psi(T) \right)  \preccurlyeq f(\mathbf{u}) \preccurlyeq g_t\left( {\pmb \xi} + s_1\psi(T) \right)  \right\rbrace \right) \mathrm{d} t \\
&\asymp_{n, \ell, f} \ \int_{M}^{s_2 T} \,  t^{n-1} \, t^{-d} \, \left( \left(2s_1\right)^\ell \, \prod_{j=1}^\ell \psi_j(T) \right) \mathrm{d} t \\
&\asymp_{\ell, s_1} \ \int_{M}^{s_2 T} \,  t^{n-(d+1)} \, \left( \prod_{j=1}^\ell \psi_j(T) \right) \mathrm{d} t \\  
&=  \left[\left(s_2 T \right)^{n-d} - M^{n-d}  \right] \left( \prod_{j=1}^\ell \psi_j(T) \right) \\
&\asymp_{n, d, s_2, M} \ T^{n-d} \, \prod_{j=1}^\ell \psi_j(T).   
\end{align*} It follows that  \[ 0 < \liminf_{T \to \infty} \frac{T^{n-d} \,  \prod_{j=1}^\ell \psi_j\left(T\right)}{m\left(B_{{_{\pmb \xi} f}, s_1\psi\left(T \right), \|\cdot\|, {s_2 T}} \right)} \leq \limsup_{T \to \infty} \frac{T^{n-d} \,  \prod_{j=1}^\ell \psi_j\left(T\right)}{m\left(B_{{_{\pmb \xi} f}, s_1\psi\left(T \right), \|\cdot\|, {s_2 T}} \right)} < \infty.  \] 

\vspace{0.025in}

\noindent Lemma \ref{scalingonly}(ii) now implies that the pair $\left({_{\pmb \xi} f}, \psi\right)$ is uniformly acceptable. It is clear from the preceding work that \begin{equation*}
\inf_{N \in \mathbb{Z}_{\geq 1}}\sum_{k=N}^{\infty} m\left( B_{{_{\pmb \xi} f}, \psi\left( 2^k \right), \|\cdot\|, {2^k}} \right)^{1-r} < \infty \quad \text{if and only if} \quad \sum_{k=1}^{\infty}\left[ 2^{k(n-d)} \prod_{j=1}^\ell \psi_j\left(2^k\right) \right]^{1-r} < \infty. \end{equation*} The desired result now follows from Theorem \ref{mainuniformresult}. \end{itemize} \qedhere\end{proof}  


We shall now consider examples of $f : \mathbb{R}^n \to \mathbb{R}$ that do not satisfy the nonsingularity hypotheses of Theorem \ref{submerapprox}. Since the measure estimates furnished by Theorem \ref{submersionvol} are no longer available in this setting, we shall instead use some \textit{ad hoc} measure calculations that were performed in the authors' previous paper:~see \cite[Corollaries 4.2 and 4.3]{KS}. In what follows, for each $i \in \mathbb{Z}_{\geq 0}$, we write $\log^i$ to denote the function $\mathbb{R}_{>0} \to \mathbb{R}$ given by $t \mapsto \left(\log t\right)^i$; in particular, $\log^0$ denotes the constant function that is equal to $1$ everywhere on $\mathbb{R}_{>0}.$ Our first example expands upon \cite[Corollary 4.2]{KS}; in that corollary, we considered the function $f : \mathbb{R}^n \to \mathbb{R}$  given by \eq{product}{f(\mathbf{x}) := \prod_{i=1}^n \left| x_i \right| } and essentially proved the following result. 

\begin{lem}\label{lemmaprodcoord}\cite[Corollary 4.2(i)]{KS} Let $f : \mathbb{R}^n \to \mathbb{R}$ be as in \equ{product}, let ${\overline{\psi}} : \mathbb{R}_{\geq 0} \to \mathbb{R}_{>0}$ be bounded and Borel measurable, and let $\eta$ denote the maximum norm on $\mathbb{R}^n.$ Then there exists $R = R \left({\overline{\psi}}, n \right) \in \mathbb{R}_{\geq 1}$ such that for every Borel measurable function $\psi : \mathbb{R}_{\geq 0} \to \mathbb{R}_{>0}$ with $\psi \leq {\overline{\psi}}$ on $\mathbb{R}_{\geq 0}$ and any real numbers $S$ and $T$ with $R \leq S \leq T,$ we have 
\begin{equation}
m\left(A_{f, \psi, \eta} \cap \{\mathbf{x} \in \mathbb{R}^n : {S} \leq\eta(\mathbf{x}) \leq {T} \} \right) = 2^n \, n \int_{S}^{T} \frac{\psi({t})}{{t}}
 \left[ \sum_{i=0}^{n-2} \frac{1}{i!} \log^i\left(\frac{{t}^n}{\psi({t})}\right)\right] \mathrm{d} t. \end{equation} 
 \end{lem}  
\noindent We remark that, strictly speaking, an application of \cite[Corollary 4.2(i)]{KS} would require the function $\psi$ to be nonincreasing (see \cite[Remark 4.4(iii)]{KS}) and would provide a value of $R$ dependent on $\psi.$ That being said, an inspection of said corollary's proof shows that only the boundedness and Borel measurability of $\psi$ are required and not its monotonicity; this inspection furthermore shows that one may choose \[ R = R\left({\overline{\psi}}, n \right) := 1 + \left( \sup\left\lbrace {\overline{\psi}}(t) : t \in \mathbb{R}_{\geq 0 }\right\rbrace  \right)^{1/n}.  \]


\smallskip

We now consider a generalized version of the function in \equ{product}, given by raising that function to an arbitrary power $\omega \in \mathbb{R}_{>0}.$ 

\begin{thm}\label{productcoord}
Let $\psi : \mathbb{R}_{\geq 0} \to \mathbb{R}_{>0}$ be  regular and nonincreasing. Let $\omega \in \mathbb{R}_{>0}$ be arbitrary. Let $f : \mathbb{R}^n \to \mathbb{R}$ be given by $f(\mathbf{x}) := \left(\prod_{i=1}^n \left| x_i \right| \right)^\omega.$ Let $\xi \in \mathbb{R}_{\geq 0}.$ Then the following hold.      
\vspace{0.05in}   
\begin{itemize}       
\item[(i)]  If \[ \begin{cases}
 { \displaystyle \int_1^{\infty} \frac{\psi(t)^{1/\omega}}{t}\log^{n-2}\left(t\right) \mathrm{d} t }  &  \text{ if } \xi = 0  \\ { \displaystyle \int_1^{\infty} \frac{\psi(t)}{t} \log^{n-2}\left(t\right) \mathrm{d} t }  &  \text{ if } \xi > 0 \end{cases} \] is finite $($respectively, infinite$)$,
then $\left({_{ \xi}} f\right) \circ g$ is $\left(\psi, \nu, \mathcal{P}\right)$-approximable for Haar-almost no $($respectively, almost every$)$ $g \in G$. 
\vspace{0.05in}  
\item[(ii)] If the infinite series \[ \begin{cases}
{ \displaystyle \sum_{k=1}^{\infty}\left[ k^{n-1}  \psi\left(2^k\right) ^{1/\omega} \right]^{1-r} }   &  \text{ if } \xi = 0  \\ { \displaystyle \sum_{k=1}^{\infty}\left[ k^{n-1} \psi\left(2^k\right) \right]^{1-r} }  &  \text{ if } \xi > 0 \end{cases} \] converges, then $\left({_{ \xi}} f\right) \circ g$ is {\sl uniformly} $\left(\psi, \nu, \mathcal{P}\right)$-approximable for Haar-almost every $g \in G$.
\end{itemize} \end{thm}  

\smallskip                

\begin{proof}
If $\displaystyle \lim_{t \to \infty} \psi(t) > 0$, then note that we can easily construct a function $\varphi : \mathbb{R}_{\geq 0} \to \mathbb{R}_{>0}$ that is regular, nonincreasing, satisfies $\displaystyle \lim_{t \to \infty} \varphi(t) = 0$, and for which the following hold:~when $\psi$ is replaced by $\varphi$ in Theorem \ref{productcoord}, the integral in (i) diverges and the infinite series in (ii) converges. The conclusions of (i) and (ii) will then follow for $\varphi$; we then infer that the conclusions of (i) and (ii) follow for $\psi$ as well. We therefore assume without loss of generality that $\displaystyle \lim_{t \to \infty} \psi(t) = 0.$ In concert with the regularity and monotonicity of $\psi$, this implies that there exists some $\lambda = \lambda(\psi) \in \mathbb{R}_{>0}$ such that for every sufficiently large $t \in \mathbb{R}_{\geq 1}$, we have \begin{equation}\label{regularpsilog}  1 \leq -\log\big(\psi(t)\big) \leq \lambda\log(t). \end{equation}  


Let us first discuss the case $\xi = 0$, which was the subject of \cite[Corollary 4.2]{KS}. Let us assume that $\omega = 1$, since the general case $\xi = 0$ reduces to this particular sub-case. Even though the integral and summatory conditions in that corollary and those in the $\xi = 0$ case of this theorem look different from one another, they are actually equivalent. Indeed, \eqref{regularpsilog} implies that the integral \[ \int_1^{\infty} \frac{\psi({t})}{{t}}  \log^{n-2} \left(\frac{{t}^n}{\psi({t})} \right) \mathrm{d}{t}\] in \cite[Corollary 4.2(ii)]{KS} converges if and only if the integral $\displaystyle \int_1^{\infty} \frac{\psi(t)}{t}\log^{n-2}\left(t\right) \mathrm{d} t $ converges. Likewise, the infinite series in \cite[Corollary 4.2(iii)]{KS}, converges if and only if the infinite series $\displaystyle \sum_{k=1}^{\infty}\left[ k^{n-1} \psi\left(2^k\right) \right]^{1-r}$ converges. The $\xi = 0$ case of this theorem then follows from the preceding work and \cite[Corollary 4.2]{KS}.  

\medskip

Suppose now that $\xi > 0.$ Let $\eta$ denote the maximum norm on $\mathbb{R}^n.$ 

\smallskip

\begin{itemize}
\item[(i)] Let $s \in \mathbb{R}_{>0}$ be given. Define $\overline{\psi} : \mathbb{R}_{\geq 0} \to \mathbb{R}_{>0}$ by $\overline{\psi}(t) := \left( \xi + s\psi \right)^{1/\omega}$. Let $R = R(\overline{\psi}, n)$ be as in Lemma \ref{lemmaprodcoord}. Fix $M \in \mathbb{R}$ for which $M > R$ and $\displaystyle s\psi(M) < \xi/2.$ Then for each $t \in \mathbb{R}_{\geq M}$, we have $\displaystyle s\psi(t) < \xi/2.$ Define $\underline{\psi} : \mathbb{R}_{\geq 0} \to \mathbb{R}_{>0}$ by \[ \underline{\psi}(t) := \begin{cases}
 { \displaystyle \overline{\psi}(t) }  &  \text{ if } t \in [0, M)  \\ { \displaystyle \left( \xi - s\psi(t)\right)^{1/\omega} }  &  \text{ if } t \in [M, \infty) \end{cases}. \] Let $h : \mathbb{R}^n \to \mathbb{R}$ be given by $h(\mathbf{x}) := \prod_{i=1}^n \left| x_i \right|.$  Note that  \begin{equation}\label{preprodsubtract}   \left( A_{{{h}}, \overline{\psi}, \eta} \smallsetminus A_{{{h}}, \underline{\psi}, \eta}\right) \cap \eta^{-1}\left(\mathbb{R}_{\geq M}\right) \subseteq A_{{_{ \xi} {f}}, s\psi, \eta} \cap \eta^{-1}\left(\mathbb{R}_{\geq M}\right)
 \end{equation} and \begin{equation}\label{prodsubtract}  
 m\Bigg( \bigg(A_{{_{ \xi} {f}}, s\psi, \eta} \cap \eta^{-1}\left(\mathbb{R}_{\geq M}\right) \bigg) \smallsetminus \bigg( \left( A_{{{h}}, \overline{\psi}, \eta} \smallsetminus A_{{{h}}, \underline{\psi}, \eta}\right) \cap \eta^{-1}\left(\mathbb{R}_{\geq M}\right) \bigg) \Bigg) = 0.  \end{equation}  Now fix any real numbers $S$ and $T$ with $M \leq S \leq T.$ Using Lemma \ref{lemmaprodcoord}, it follows that \begin{equation}\label{prodover} m\left( A_{{{h}}, \overline{\psi}, \eta}  \cap \left\lbrace \mathbf{x} \in \mathbb{R}^n : S \leq \eta(\mathbf{x})  \leq T \right\rbrace \right) =    2^n \, n \int_{S}^{T} \frac{\big( \xi + s\psi(t)\big)^{1/\omega}}{{t}}
 \left[ \sum_{i=0}^{n-2} \frac{1}{i!} \log^i\left(\frac{{t}^n}{\big( \xi + s\psi(t)\big)^{1/\omega}}\right)\right] \mathrm{d} t
\end{equation} and  \begin{equation}\label{produnder}  m\left( A_{{{h}}, \underline{\psi}, \eta}  \cap \left\lbrace \mathbf{x} \in \mathbb{R}^n : S \leq \eta(\mathbf{x})  \leq T \right\rbrace \right) 
 =    2^n \, n \int_{S}^{T} \frac{\big( \xi - s\psi(t)\big)^{1/\omega}}{{t}}
 \left[ \sum_{i=0}^{n-2} \frac{1}{i!} \log^i\left(\frac{{t}^n}{\big( \xi - s\psi(t)\big)^{1/\omega}}\right)\right] \mathrm{d} t.  
\end{equation} 
It is easy to see that \begin{equation}\label{integralconditionone}
 m\left( A_{{_{ \xi} f}, s\psi, \eta}\right) = \infty \quad \quad \quad \quad \text{if and only if} \quad \quad \quad \quad \int_{1}^{\infty} \frac{\psi(t)}{t} \log^{n-2}\left(t\right) \mathrm{d} t = \infty. 
\end{equation} This follows from \eqref{preprodsubtract} and \eqref{prodsubtract}, subtracting the right-hand side of \eqref{produnder} from that of \eqref{prodover}, using \eqref{regularpsilog} and the dominance of $\log^{n-2}$, and then performing two first-order Taylor approximations. Note that the integral criterion in \eqref{integralconditionone} is independent of $s \in \mathbb{R}_{>0}.$ Lemma \ref{scalingonly}(i) then implies that the pair $\left({_{ \xi} f}, \psi\right)$ is asymptotically acceptable. The desired result now follows from the foregoing work and Theorem \ref{asymptoticmain}.   

\vspace{0.05in}

\item[(ii)] Let $s_1, s_2 \in \mathbb{R}_{>0}$ be given. Set $J := 1 + \psi(0).$ Arguing as in part (i), we infer that there exists $M \in \mathbb{R}_{> 2}$ such that for each $T \in \mathbb{R}$ with $T > M/s_2$, we have 
\begin{align*}
m\left( \left\lbrace \mathbf{x} \in B_{{_{ \xi} f}, {s_1 \psi(T)}, \eta, {s_2 T}} :  \eta(\mathbf{x}) \geq M \right\rbrace \right)
&\asymp_{n, J, M, s_1, s_2} \, \int_M^{s_2 T} \psi(T) \ t^{-1} \, \log^{n-2}(t) \, \mathrm{d} t  \\
\vspace{0.025in}  
&= \psi(T) \, \left[ \log^{n-1}\left(s_2 T \right) - \log^{n-1}(M) \right] \\
\vspace{0.025in}   
&
\asymp_{n, M, s_2} \, \psi(T) \, \log^{n-1}(T). 
\end{align*} 
It follows that \[ 0 < \liminf_{T \to \infty} \frac{ \psi(T) \, \log^{n-1}(T) }{m\left(B_{{_{\xi} f}, s_1\psi(T), \eta, {s_2 T}} \right)} \leq \limsup_{T \to \infty} \frac{ \psi(T) \, \log^{n-1}(T) }{m\left(B_{{_{\xi} f}, s_1\psi(T), \eta, {s_2 T}} \right)} < \infty.  \] Lemma \ref{scalingonly}(ii) then implies that the pair $\left({_{ \xi} f}, \psi\right)$ is uniformly acceptable. It is clear from the foregoing work that \begin{equation*}
\inf_{N \in \mathbb{Z}_{\geq 1}}\sum_{k=N}^{\infty} m\left( B_{{_{\xi} f}, \psi\left( 2^k \right), \eta, {2^k}} \right)^{1-r} < \infty \quad \text{if and only if} \quad \sum_{k=1}^{\infty}\left[ k^{n-1} \psi\left(2^k\right) \right]^{1-r} < \infty. \end{equation*} 
{An application of Theorem \ref{mainuniformresult} then yields the desired result.}
\end{itemize} 
 \end{proof}       

\begin{rmk} \rm { It is easy to see that one may modify the preceding proof to obtain a similar result when $\xi \in \mathbb{R}$ and $f : \mathbb{R}^n \to \mathbb{R}$ is a function of the form  $f(\mathbf{x}) := \left(\prod_{i=1}^n x_i \right)^{q_1/q_2},$ where  $q_1$ and $q_2$ are arbitrary odd natural numbers. } 
\end{rmk}




Our next and final example expands upon \cite[Corollary 4.3]{KS} and is of interest because of its relation to the Khintchine\textendash Groshev
Theorem.


\begin{thm}\label{khintchine} Let $\psi : \mathbb{R}_{\geq 0} \to \mathbb{R}_{>0}$ be  regular and nonincreasing. Let $p \in \{1, \dots , n - 1 \}$ and $\mathbf{z} = (z_1, \dots , z_p) \in \left(\mathbb{R}_{>0}\right)^p$ be given. Let $f: \mathbb{R}^n \to \mathbb{R}$ be given by \[ f(x_1, \dots , x_n) := \max\left\lbrace |x_i|^{z_i}: 1 \leq i \leq p\right\rbrace. \] Set $z := \sum_{i=1}^p  \left(z_i\right)^{-1}.$ Let $\xi \in \mathbb{R}_{\geq 0}.$ Then the following hold.    
\vspace{0.05in}     
\begin{itemize}
\item[ \rm (i)] If \[ \begin{cases}
 { \displaystyle \int_1^{\infty} \psi(t)^z  {t}^{n - (p+1)} \ \mathrm{d} t  }  &  \text{ if } \xi = 0  \\ { \displaystyle \int_1^{\infty} \psi(t) \, {t}^{n - (p+1)} \ \mathrm{d} t  }  &  \text{ if } \xi > 0 \end{cases} \] is finite $($respectively, infinite$)$,
then $\left({_{\xi}} f\right) \circ g$ is $\left(\psi, \nu, \mathcal{P} \right)$-approximable for Haar-almost no $($respectively, almost every$)$ $g \in G.$
\vspace{0.05in}
\item[\rm (ii)] If the infinite series \[ \begin{cases}
 { \displaystyle \sum_{k = 1}^\infty 
      \left[ 2^{(n-p)k} \psi\left(2^k\right) ^z \right]^{1-r}}   &  \text{ if } \xi = 0  \\ { \displaystyle \sum_{k = 1}^\infty 
      \left[ 2^{(n-p)k} \, \psi\left(2^k\right) \right]^{1-r}}  &  \text{ if } \xi > 0 \end{cases} \] converges, then $\left({_{\xi}} f\right) \circ g$ is {\sl uniformly} $\left(\psi, \nu, \mathcal{P} \right)$-approximable for Haar-almost every $g \in G.$ \end{itemize}  \end{thm} 

\noindent  The proof of this theorem makes use of the following lemma.    


\begin{lem}\cite[Corollary 4.3(i)]{KS}\label{lemmakhintchine} Let $p \in \{1, \dots , n - 1 \}$ and $\mathbf{z} = (z_1, \dots , z_p) \in \left(\mathbb{R}_{>0}\right)^p$ be given. Let $f: \mathbb{R}^n \to \mathbb{R}$ be given by \[f(x_1, \dots , x_n) := \max\left\lbrace |x_i|^{z_i}: 1 \leq i \leq p\right\rbrace.\] Set $z := \sum_{i=1}^p  \left(z_i\right)^{-1}.$ Let ${\overline{\psi}} : \mathbb{R}_{\geq 0} \to \mathbb{R}_{>0}$ be bounded and Borel measurable. Let $\eta$ denote the maximum norm on $\mathbb{R}^n.$ Then there exists some $R = R\left({\overline{\psi}}, \mathbf{z} \right) \in \mathbb{R}_{\geq 1}$ such that for every Borel measurable function $\psi : \mathbb{R}_{\geq 0} \to \mathbb{R}_{>0}$ with $\psi \leq {\overline{\psi}}$ on $\mathbb{R}_{\geq 0}$ and any real numbers $S$ and $T$ with $R \leq S \leq T,$ we have \begin{equation*}\label{lemmaresult}
m\left(A_{f, \psi, \eta} \cap \{\mathbf{x} \in \mathbb{R}^n : {S} \leq\eta(\mathbf{x}) \leq {T} \} \right) = 2^n (n-p) \int_{S}^{T} \psi({t})^z \, t^{n-(p+1)} \, \mathrm{d} t. 
\end{equation*}
\end{lem}   

\noindent  The relationship between Lemma \ref{lemmakhintchine} and \cite[Corollary 4.3(i)]{KS} is analogous to that between Lemma \ref{lemmaprodcoord} and \cite[Corollary 4.2(i)]{KS}, and similar remarks to those made earlier apply here. In particular, inspecting the proof of \cite[Corollary 4.3(i)]{KS} shows that one may choose \[  R = R\left({\overline{\psi}}, \mathbf{z} \right) := 1 + \max_{1 \leq i \leq p}  \left(\sup\left\lbrace {\overline{\psi}}(t) : t \in \mathbb{R}_{\geq 0 }\right\rbrace\right)^{1/{z_i}} \] in Lemma \ref{lemmakhintchine}. The $\xi = 0$ case of  Theorem \ref{khintchine} is already known:~see \cite[Corollary 4.3]{KS}. The proof of the $\xi > 0$ case of this theorem is similar to, and simpler than, that of the $\xi > 0$ case of Theorem \ref{productcoord}:~it is therefore omitted.  

\smallskip   

\begin{rmk}\label{follow} \rm 
{Let us mention that all the results in \S1 (in particular, Theorems \ref{simplesubmerapprox}, \ref{simpleproduct}, and  \ref{simplemax} and Remarks \ref{primasl} and \ref{singularprimasl}) follow from those here in \S3 and the fact that   $G = \mathrm{SL}_n(\mathbb{R})$ and $G = \mathrm{ASL}_n(\mathbb{R})$ satisfy various forms of the Siegel-type and Rogers-type axioms:~see \cite[Theorems 2.5, 2.6, and 2.8]{KS} and the references therein for details. }  \end{rmk}


\ignore{ \begin{proof}[Proof of Theorem \ref{khintchine}] Note that \cite[Corollary 4.3]{KS} implies both the $\xi = 0$ case of (i) and that of (ii).  

\medskip 

Suppose now that $\xi > 0.$ Arguing as in the proof of Theorem \ref{productcoord}, we assume without loss of generality that $\displaystyle \lim_{t \to \infty} \psi(t) = 0.$ Define ${\overline{\psi}} : \mathbb{R}_{\geq 0} \to \mathbb{R}_{>0}$ to be the constant function equal to $\displaystyle 2 \xi + \sup\left\lbrace \psi(t) : t \in \mathbb{R}_{\geq 0}\right\rbrace$. Let $R = R\left({\overline{\psi}}, \mathbf{z} \right)$ be as in Lemma \ref{lemmakhintchine}. Let $a = a(\psi) \in \mathbb{R}_{>1}$ and $b = b(\psi) \in \mathbb{R}_{>0}$ be as in Definition \ref{basicapdefns}. Fix $i \in \mathbb{Z}_{\geq 1}$ for which $a^i > 3.$ Let $c \in \mathbb{R}_{>0}$ be arbitrary. Fix $M \in \mathbb{R}_{> R}$ for which $\displaystyle \psi(M) < \frac{\xi}{4} \cdot \min \left\lbrace c, c^{-1} \right\rbrace$. 



If $\displaystyle m\left(A_{ _{  \xi} f, c\psi, \nu}\right) < \infty,$ then $\displaystyle \sum_{k=1}^{\infty} m\left(\{ \mathbf{x} \in A_{ _{  \xi} f, c\psi({k+1}), \nu} : k \leq \nu(\mathbf{x}) \leq {k+1} \}\right) < \infty$ because the function $c\psi$ is nonincreasing. For each $k \in \mathbb{Z}_{> M}$, we then have  \begin{align*} m\left(\{ \mathbf{x} \in A_{ _{  \xi} f, c\psi({k+1}), \nu} : k \leq \nu(\mathbf{x}) \leq {k+1} \}\right) &\asymp_{n, p, \nu, \xi, z} \, c\psi(k+1) \int_{k}^{k+1} t^{n-(p+1)} \, \mathrm{d} t \\
&\geq c \, b^i \psi\left(a^{-i}(k+1)\right)\int_{k}^{k+1} t^{n-(p+1)} \, \mathrm{d} t \\
&\geq c \, b^i \int_{k}^{k+1} \psi(t) \, t^{n-(p+1)} \, \mathrm{d} t.  \end{align*} It now follows that if $\displaystyle m\left(A_{ _{  \xi} f, c\psi, \nu}\right) < \infty,$ then $\displaystyle \int_1^{\infty} \psi(t) \, t^{n-(p+1)} \, \mathrm{d} t < \infty.$    


\smallskip


If $\displaystyle m\left(A_{ _{  \xi} f, c\psi, \nu}\right) = \infty,$ then $\displaystyle \sum_{k=1}^{\infty} m\left(\{ \mathbf{x} \in A_{ _{  \xi} f, c\psi(k), \nu} : k \leq \nu(\mathbf{x}) \leq {k+1} \}\right) = \infty.$ For each $k \in \mathbb{Z}_{> M},$ we then have \begin{align*} m\left(\{ \mathbf{x} \in A_{ _{  \xi} f, c\psi({k}), \nu} : k \leq \nu(\mathbf{x}) \leq {k+1} \}\right) &\asymp_{n, p, \nu, \xi, z} \, c\psi(k) \int_{k}^{k+1} t^{n-(p+1)} \, \mathrm{d} t \\
&\leq c \, b^{-i} \psi\left(a^i k\right) \int_{k}^{k+1} t^{n-(p+1)} \, \mathrm{d} t \\
&\leq c \, b^{-i} \int_{k}^{k+1} \psi(t) \, t^{n-(p+1)} \, \mathrm{d} t.  
\end{align*} It now follows that if $\displaystyle m\left(A_{ _{  \xi} f, c\psi, \nu}\right) = \infty,$ then $\displaystyle \int_1^{\infty} \psi(t) \, t^{n-(p+1)} \, \mathrm{d} t = \infty.$ 


Hence, \[  m\left(A_{ _{  \xi} f, c\psi, \nu}\right) < \infty \quad \quad \quad \text{if and only if} \quad \quad \quad \int_1^{\infty} \psi(t) \, t^{n-(p+1)} \, \mathrm{d} t < \infty. \] Since $c \in \mathbb{R}_{>0}$ is arbitrary and $\nu$ is an arbitrary norm on $\mathbb{R}^n,$ we thus infer that the pair $\left({_{ \xi} f}, \psi\right)$ is asymptotically acceptable. Statement (i) now follows from the preceding work and Theorem \ref{asymptoticmain}.    

\vspace{0.1in}

The proof of (ii) is very similar to that of (i) and previous proofs and is therefore omitted.  \end{proof} }




\begin{rmk} \rm
We note here that one may easily deduce analogues of
{the statements made in Remark \ref{examples}}
for Theorems \ref{productcoord},
\ref{khintchine}, and for the $\ell = 1$ case of Theorem \ref{submerapprox}. We also note that the $\ell = 1$ case of Theorem \ref{submerapprox} applies to the functions discussed in {Remark \ref{examples}}. \end{rmk} 

\begin{rmk} \rm
In light of the preceding results, we note that the discussion in Remark \ref{simplezerovsnonzero} applies in the current, more general, setting.    
\end{rmk}



\subsection*{Acknowledgements}
The co-author Skenderi would like to thank Jayadev Athreya and Jon Chaika for various discussions.      


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\end{bibdiv}

\end{document}


 \ignore{\begin{align*}
    m\left( A_{{_{ \xi} {f}}, c\psi, \eta} \cap \left\lbrace \mathbf{x} \in \mathbb{R}^n : S \leq \eta(\mathbf{x})  \leq T \right\rbrace \right) \\
 =    2^n \, n \int_{S}^{T}\left( \frac{\big( \xi + c\psi(t)\big)^{1/\omega}}{{t}}
 \left[ \sum_{i=0}^{n-2} \frac{1}{i!} \log^i\left(\frac{{t}^n}{\big( \xi + c\psi(t)\big)^{1/\omega}}\right)\right] - \frac{\big( \xi - c\psi(t)\big)^{1/\omega}}{{t}}
 \left[ \sum_{i=0}^{n-2} \frac{1}{i!} \log^i\left(\frac{{t}^n}{\big( \xi - c\psi(t)\big)^{1/\omega}}\right)\right] \right)\, \mathrm{d} t\\
    &\asymp_{n, \xi, \theta} \, \int_{S}^{T} \frac{\big( \xi + c\psi(t) \big) - \big(\xi - c\psi(t) \big)}{t} \log^{n-2}\left(t\right) \mathrm{d} t \\
    &\asymp_{c} \, \int_{S}^{T} \frac{\psi(t)}{t} \log^{n-2}\left(t\right) \mathrm{d} t. 
\end{align*} 

 Since $\xi > 0$ and $\psi$ is regular, nonincreasing, and
{tends to $0$ as $t\to \infty$}, it follows that there exist $\theta = \theta(\xi, \psi, c) \in \mathbb{Z}_{\geq 1}$ and $M_1 = M_1(n, \xi, \psi, c) \in \mathbb{R}_{\geq 1}$ such that for each $t \in \mathbb{R}_{\geq M_1}$ we have 
\begin{equation}\label{boundthetan}
0 < t^{-\theta} < c\psi(t) < \xi - c\psi(t)  < \xi + c\psi(t) < t. 
\end{equation}



\smallskip



\noindent Now set $M := 2 + \max\big\{ M_1, 2\big( \xi + c \, \psi(0) \big) \big\}.$ Let $\varphi$
denote either one of the functions $\mathbb{R}_{\geq 0} \to \mathbb{R}$ that is given by $\xi - c\psi(\cdot)$ or $\xi + c\psi(\cdot).$ It now follows from \cite[Corollary 4.2]{KS} and its proof and \eqref{boundthetan} \comm{(We need more detail here, both about the uniformity you mentioned, and about the fact that $\phi$ is not non-increasing -- will the proof still work? I think it should. Maybe another lemma?)} that for any $S \in \mathbb{R}$ and any $T \in \mathbb{R}$ with $M \leq S \leq T$, we have \begin{align*} m\left(A_{{f}, \varphi, \eta} \cap \left\lbrace \mathbf{x} \in \mathbb{R}^n : S \leq \eta(\mathbf{x}) \leq T \right\rbrace \right) &= 2^n \, n \int_{S}^{T} \frac{\varphi(t)}{{t}}
 { \left[ \sum_{i=0}^{n-2} \frac{1}{i!} \log^i\left(\frac{{t}^n}{\varphi(t)}\right)\right]}
   \, \mathrm{d} t
   \\
   &\asymp_{n, \xi, \theta} \, \int_{S}^{T} \frac{\varphi(t)}{t} \log^{n-2}\left(t\right) 
   \, \mathrm{d} t.
\end{align*}
{Hence} for any $S$ and $T$ as above  we have
 \begin{align*}
    m\left( A_{{_{ \xi} {f}}, c\psi, \eta} \cap \left\lbrace \mathbf{x} \in \mathbb{R}^n : S \leq \eta(\mathbf{x}) \leq T \right\rbrace \right) &\asymp_{n, \xi, \theta} \, \int_{S}^{T} \frac{\big( \xi + c\psi(t) \big) - \big(\xi - c\psi(t) \big)}{t} \log^{n-2}\left(t\right) \mathrm{d} t \\
    &\asymp_{c} \, \int_{S}^{T} \frac{\psi(t)}{t} \log^{n-2}\left(t\right) \mathrm{d} t. 
\end{align*} 
\comm{The last step is not clear -- do you mean that the implicit constant does not depend on $\psi$? $\theta$ surely does...}
Fix $\kappa \in \mathbb{Z}_{\geq 1}$ for which $a^{-\kappa} \, \eta \leq \nu \leq a^\kappa \, \eta.$ Then for each $\mathbf{x} \in \mathbb{R}^n,$ we have \[  b^{\kappa} \, \psi\big(\eta(\mathbf{x})\big) \preccurlyeq \psi\big(\nu(\mathbf{x})\big) \preccurlyeq b^{-\kappa} \, \psi\big(\eta(\mathbf{x})\big). \] {Therefore} for any norm ${\nu}$ on $\mathbb{R}^n$ and any ${c} \in \mathbb{R}_{>0}$, 
\new{we obtain}
 \[ m\left(A_{{_{ \xi} f}, \new{c}\psi, \new{\nu}}\right) < \infty \quad \quad \text{if and only if} \quad \quad \int_1^{\infty} \frac{\psi(t)}{t} \log^{n-2}\left(t\right) \mathrm{d} t < \infty \] 

 In particular,
 $\left({_{ \xi} f}, \psi\right)$
 is asymptotically acceptable. The foregoing work and Theorem \ref{asymptoticmain} then yield (i) in its entirety. } 


Let $c_1, c_2, c_3 \in \mathbb{R}_{>0}$ be given. Define $\displaystyle \gamma = \gamma\left(\xi, c_1\right) := \frac{\xi}{2  c_1}  > 0.$ Arguing as in part (i), it follows that there exist $\theta = \theta(\xi, \psi, c_1) \in \mathbb{Z}_{\geq 1}$ and $M_1 = M_1(n, \xi, \psi, c_1) \in \mathbb{R}_{\geq 1}$ \comm{(are these the same as in (i)? then we can simply say so)} such that for every $\varepsilon \in (0, \gamma) \subset \mathbb{R}$ and every $t \in \mathbb{R}_{\geq M_1},$ we have  \begin{equation}\label{firstuniform}
0 < t^{-1} < \frac{\xi}{2} < \xi - c_1\varepsilon < \xi + c_1\varepsilon < \frac{3\xi}{2} < t \end{equation} and \begin{equation}\label{seconduniform}  0 < \psi(t) \cdot \max\{1, c_1 \} < \gamma.    \end{equation}  Appealing to \cite[Corollary 4.2]{KS} and its proof and \eqref{firstuniform} and arguing as in part (i) \comm{(need to make changes here as well)}, we infer the following:~there exists some $M \in \mathbb{R}_{\geq M_1}$ such that for every $\varepsilon \in (0, \gamma)$ and every $T \in \left[\frac{c_2M}{c_3}, \infty\right),$ we have  \begin{align*}
m\left( \left\lbrace \mathbf{x} \in \mathbb{R}^n : \left|{_{ \xi} \new{f}}(\mathbf{x}) \right| \leq c_1 \, \varepsilon \ \ \text{and} \ \ c_2 \, M \leq \eta(\mathbf{x}) \leq c_3 \, T \right\rbrace \right) &\asymp_{n, \xi, c_1} \, \int_{c_2M}^{c_3T} \frac{c_1 \varepsilon}{t} \log^{n-2}\left(t\right) \mathrm{d} t \\
&{\asymp_{n, \xi, c_1, c_2, c_3, M}} \ \ \varepsilon \,  \log^{n-1}(T).   
\end{align*} It follows from \eqref{seconduniform} and the preceding work that for every $T \in \mathbb{R}$ with $T > \max\left\lbrace M, \frac{c_2M}{c_3} \right\rbrace$, we have \eq{useful2}{m\left( \left\lbrace \mathbf{x} \in \mathbb{R}^n : \left|{_{ \xi} \new{f}}(\mathbf{x}) \right| \leq c_1 \, \psi\left(T\right) \ \ \text{and} \ \ c_2 \, M \leq \eta(\mathbf{x}) \leq c_3 \, T \right\rbrace \right) {\asymp_{n, \xi, c_1, c_2, c_3, M}} \ \ \psi\left(T\right)  \log^{n-1}(T). } Since $c_1, c_2, c_3 \in \mathbb{R}_{>0}$ are arbitrary and all norms on $\mathbb{R}^n$ are equivalent, it is easy to see that \equ{useful2} implies the following:~for any $s_1, s_2 \in \mathbb{R}_{>0},$ we have \eq{acceptableprod1}{ 0 < \liminf_{T \to \infty} \frac{\psi\left(T\right)  \log^{n-1}(T)}{m\left(B_{{_{ \xi} \new{f}}, s_1\psi\left(T \right), \nu, {s_2 T}} \right)} \leq \limsup_{T \to \infty} \frac{\psi\left(T\right)  \log^{n-1}(T)}{m\left(B_{{_{ \xi} \new{f}}, s_1\psi\left(T \right), \nu, {s_2 T}} \right)} < \infty. } Therefore, the pair $\left({_{ \xi} \new{f}}, \psi\right)$ is uniformly acceptable. Let $u_\bullet = (u_k)_{k \in \mathbb{Z}_{\geq 1}}$ is any strictly increasing sequence of elements of $\mathbb{R}_{>1}$ with $\displaystyle \lim_{k \to \infty} u_k = \infty.$ Then \equ{acceptableprod1} implies  \begin{equation}\label{uniformprodeq}
\inf_{N \in \mathbb{Z}_{\geq 1}}\sum_{k=N}^{\infty}  m\left(B_{{_{ \xi} \new{f}}, \psi\left(u_k\right), \nu, u_k}\right)^{1-r} < \infty \quad \text{if and only if} \quad \sum_{k=1}^{\infty} \left[\psi\left(u_k\right)\log^{n-1}\left(u_k\right) \right]^{1-r} < \infty. \end{equation}


Statement (ii) now follows in its entirety from the foregoing work, the preceding observation, \eqref{uniformprodeq}, and Theorem \ref{mainuniformresult}. 





\ignore{The following result is a variant of the preceding one; its proof is omitted because it is very similar to the proof that was just given. In what follows, recall that $\mathcal{Q}$ denotes the set of all $q \in \mathbb{Q}_{>0}$ for which there exist $q_1, q_2 \in \mathbb{Z}_{\geq 1}$ such that $q_1$ is odd, $q_2$ is odd, and $q = q_1/q_2.$ 

\begin{thm}\label{nobarsproduct}
Let $\psi : \mathbb{R}_{\geq 0} \to \mathbb{R}_{>0}$ be  regular and nonincreasing, and suppose that $\displaystyle \lim_{t \to \infty} \psi(t) = 0.$ Let $q \in \mathcal{Q}$ be arbitrary. Let $f : \mathbb{R}^n \to \mathbb{R}$ be given by $f(\mathbf{x}) := \left(\prod_{j=1}^n x_j \right)^q.$ Let $\xi \in \mathbb{R}.$ Then the following hold.   
\vspace{0.05in}
\begin{itemize}
\item[ \rm (i)]  If \[ \begin{cases}
 { \displaystyle \int_1^{\infty} \frac{(\psi(t))^{1/q}}{t} \left(\log t\right)^{n-2} \, \mathrm{d} t }  &  \text{ if } \xi = 0  \\ { \displaystyle \int_1^{\infty} \frac{\psi(t)}{t} \left(\log t\right)^{n-2} \, \mathrm{d} t }  &  \text{ if } \xi \neq 0 \end{cases} \] is finite $($respectively, infinite$)$,
then $\left({_{\xi}} f\right) \circ g$ is $\left(\psi, \nu, \mathcal{P}\right)$-approximable for Haar-almost no $($respectively, almost every$)$ $g \in G$. \vspace{0.05in}   
\item[\rm (ii)]  If the infinite series \[ \begin{cases}
{ \displaystyle \sum_{k=1}^{\infty}\left[ k^{n-1} \left( \psi\left(2^k\right) \right)^{1/q} \right]^{1-r} }   &  \text{ if } \xi = 0  \\ { \displaystyle \sum_{k=1}^{\infty}\left[ k^{n-1} \psi\left(2^k\right) \right]^{1-r} }  &  \text{ if } \xi \neq 0 \end{cases} \]  converges, then $\left({_{\xi}} f\right) \circ g$ is {\sl uniformly} $\left(\psi, \nu, \mathcal{P}\right)$-approximable for Haar-almost every $g \in G$.
\end{itemize}
\end{thm}  }




\begin{lem}  Let $p \in \{1, \dots , n - 1 \}$ and $\mathbf{z} = (z_1, \dots , z_p) \in \left(\mathbb{R}_{>0}\right)^p$ be given. Let $f: \mathbb{R}^n \to \mathbb{R}$ be given by \[f(x_1, \dots , x_n) := \max\left\lbrace |x_i|^{z_i}: 1 \leq i \leq p\right\rbrace.\] Set $z := \sum_{i=1}^p  \left(z_i\right)^{-1}.$ Let $\xi \in \mathbb{R}_{> 0}.$ Then there exists $N = N\left(\xi, \mathbf{z}, \nu\right) \in \mathbb{Z}_{\geq 1}$ such that for any real numbers $S$ and $T$ with $N \leq S \leq T$ and any $\varepsilon \in \left(0, \frac{\xi}{2}\right) \subset \mathbb{R},$ we have  \begin{equation*}\label{lemmaresult}
m\left(\left\lbrace \mathbf{x} \in \mathbb{R}^n : \left|f(\mathbf{x}) - \xi\right| \leq \varepsilon \ \text{and} \ S \leq \nu(\mathbf{x}) \leq T \right\rbrace\right) \asymp_{n, p, \nu, \xi, z} \, \varepsilon \int_{S}^{T} t^{n-(p+1)} \, \mathrm{d} t. \end{equation*}
\end{lem}

\begin{proof}
Let $\eta$ denote the maximum norm on $\mathbb{R}^n.$ If $\psi : \mathbb{R}_{\geq 0} \to \mathbb{R}_{>0}$ is any regular and nonincreasing function and $N_1 = N_1(\psi, \mathbf{z})$ is any element of $\mathbb{Z}_{\geq 1}$ for which $\displaystyle \max\left\lbrace \psi\left(N_1\right)^{1/z_i} : i \in \{1, \dots , p \} \right\rbrace < N_1$, then it follows from \cite[Corollary 4.3(i)]{KS} and its proof that for any real numbers $S_1$ and $T_1$ with $N_1 \leq S_1 \leq T_1,$ we have  \[ m\left(\left\lbrace \mathbf{x} \in \mathbb{R}^n :
f(\mathbf{x}
 \leq \psi\big(\eta(\mathbf{x})\big) \ \text{and} \ S_1 \leq \eta(\mathbf{x}) \leq T_1 \right\rbrace\right) = 2^n \, (n-p) \int_{S_1}^{T_1}  \psi(t)^z {t}^{n-(p+1)} \, \mathrm{d} t. \] Therefore, there exists $N_2 = N_2(\xi, \mathbf{z}) \in \mathbb{Z}_{\geq 1}$ such that for any real numbers $S_2$ and $T_2$ with $N_2 \leq S_2 \leq T_2$ and any $\theta \in \left[ \frac{\xi}{2}, \frac{3\xi}{2}  \right] \subset \mathbb{R}$, we have \begin{equation*}
m\left(\left\lbrace \mathbf{x} \in \mathbb{R}^n :
f(\mathbf{x}
 \leq \theta \ \text{and} \ S_2 \leq \eta(\mathbf{x}) \leq T_2 \right\rbrace\right) = 2^n \, (n-p) \, \theta^z  \int_{S_2}^{T_2}  {t}^{n-(p+1)} \, \mathrm{d} t.
\end{equation*} Since all norms on $\mathbb{R}^n$ are equivalent, there exists $N = N\left(\xi, \mathbf{z}, \nu\right) \in \mathbb{Z}_{\geq 1}$ such that for any real numbers $S$ and $T$ with $N \leq S \leq T$ and any $\theta \in \left[ \frac{\xi}{2}, \frac{3\xi}{2}  \right] \subset \mathbb{R}$, we have \begin{equation}\label{thetaxi}
m\left(\left\lbrace \mathbf{x} \in \mathbb{R}^n :
f(\mathbf{x}
 \leq \theta \ \text{and} \ S \leq \nu(\mathbf{x}) \leq T \right\rbrace\right) \asymp_{n, p, \nu} \, \theta^z  \int_{S}^{T}  {t}^{n-(p+1)} \, \mathrm{d} t.
\end{equation} 


\noindent Now let $\varepsilon \in \left(0, \frac{\xi}{2}\right) \subset \mathbb{R}$ be given. Applying \eqref{thetaxi} with $\theta = \xi + \varepsilon$ and $\theta = \xi - \varepsilon$ and then subtracting the latter resulting quantity from the former one implies that for any $S$ and $T$ as above, we have \begin{equation*}
m\left(\left\lbrace \mathbf{x} \in \mathbb{R}^n : \left|f(\mathbf{x}) - \xi\right| \leq \varepsilon \ \text{and} \ S \leq \nu(\mathbf{x}) \leq T \right\rbrace\right) \asymp_{n, p, \nu} \, \left[ \left(\xi + \varepsilon\right)^z - \left(\xi - \varepsilon\right)^z \right] \int_{S}^{T} t^{n-(p+1)} \, \mathrm{d} t. 
\end{equation*} Since $\displaystyle \left(\xi + \varepsilon\right)^z - \left(\xi - \varepsilon\right)^z \asymp_{\xi, z} \, \varepsilon$, the desired result
 follows. 
\end{proof}


\smallskip   


\begin{proof}[Proof of Theorem \ref{khintchine}] Note that \cite[Corollary 4.3]{KS} implies both the $\xi = 0$ case of (i) and that of (ii).  



Suppose now that $\xi > 0.$ Arguing as in the proof of Theorem \ref{productcoord}, we assume without loss of generality that $\displaystyle \lim_{t \to \infty} \psi(t) = 0.$ Let $N := N\left(\xi, \mathbf{z}, \nu \right) \in \mathbb{Z}_{\geq 1}$ be as in Lemma \ref{lemmakhintchine}. Let $a = a(\psi) \in \mathbb{R}_{>1}$ and $b = b(\psi) \in \mathbb{R}_{>0}$ be as in Definition \ref{basicapdefns}. Fix $i \in \mathbb{Z}_{\geq 1}$ for which $a^i > 3.$ Let $c \in \mathbb{R}_{>0}$ be arbitrary. Fix $M \in \mathbb{Z}_{\geq 1}$ for which $\displaystyle \psi(M) < \frac{\xi}{4} \cdot \min \left\lbrace c, c^{-1} \right\rbrace$. 



If $\displaystyle m\left(A_{ _{  \xi} f, c\psi, \nu}\right) < \infty,$ then $\displaystyle \sum_{k=1}^{\infty} m\left(\{ \mathbf{x} \in A_{ _{  \xi} f, c\psi({k+1}), \nu} : k \leq \nu(\mathbf{x}) \leq {k+1} \}\right) < \infty$ because the function $c\psi$ is nonincreasing. For each $k \in \mathbb{Z}_{> M}$, we then have  \begin{align*} m\left(\{ \mathbf{x} \in A_{ _{  \xi} f, c\psi({k+1}), \nu} : k \leq \nu(\mathbf{x}) \leq {k+1} \}\right) &\asymp_{n, p, \nu, \xi, z} \, c\psi(k+1) \int_{k}^{k+1} t^{n-(p+1)} \, \mathrm{d} t \\
&\geq c \, b^i \psi\left(a^{-i}(k+1)\right)\int_{k}^{k+1} t^{n-(p+1)} \, \mathrm{d} t \\
&\geq c \, b^i \int_{k}^{k+1} \psi(t) \, t^{n-(p+1)} \, \mathrm{d} t.  \end{align*} It now follows that if $\displaystyle m\left(A_{ _{  \xi} f, c\psi, \nu}\right) < \infty,$ then $\displaystyle \int_1^{\infty} \psi(t) \, t^{n-(p+1)} \, \mathrm{d} t < \infty.$    


\smallskip


If $\displaystyle m\left(A_{ _{  \xi} f, c\psi, \nu}\right) = \infty,$ then $\displaystyle \sum_{k=1}^{\infty} m\left(\{ \mathbf{x} \in A_{ _{  \xi} f, c\psi(k), \nu} : k \leq \nu(\mathbf{x}) \leq {k+1} \}\right) = \infty.$ For each $k \in \mathbb{Z}_{> M},$ we then have \begin{align*} m\left(\{ \mathbf{x} \in A_{ _{  \xi} f, c\psi({k}), \nu} : k \leq \nu(\mathbf{x}) \leq {k+1} \}\right) &\asymp_{n, p, \nu, \xi, z} \, c\psi(k) \int_{k}^{k+1} t^{n-(p+1)} \, \mathrm{d} t \\
&\leq c \, b^{-i} \psi\left(a^i k\right) \int_{k}^{k+1} t^{n-(p+1)} \, \mathrm{d} t \\
&\leq c \, b^{-i} \int_{k}^{k+1} \psi(t) \, t^{n-(p+1)} \, \mathrm{d} t.  
\end{align*} It now follows that if $\displaystyle m\left(A_{ _{  \xi} f, c\psi, \nu}\right) = \infty,$ then $\displaystyle \int_1^{\infty} \psi(t) \, t^{n-(p+1)} \, \mathrm{d} t = \infty.$ 


Hence, \[  m\left(A_{ _{  \xi} f, c\psi, \nu}\right) < \infty \quad \quad \quad \text{if and only if} \quad \quad \quad \int_1^{\infty} \psi(t) \, t^{n-(p+1)} \, \mathrm{d} t < \infty. \] Since $c \in \mathbb{R}_{>0}$ is arbitrary and $\nu$ is an arbitrary norm on $\mathbb{R}^n,$ we thus infer that the pair $\left({_{ \xi} f}, \psi\right)$ is asymptotically acceptable. Statement (i) now follows from the preceding work and Theorem \ref{asymptoticmain}.    

\vspace{0.1in}

The proof of (ii) is very similar to that of (i) and previous proofs and is therefore omitted.  \end{proof}











\begin{lem}\label{lemmaprodcoord}\cite[Corollary 4.2(i)]{KS} Let $f : \mathbb{R}^n \to \mathbb{R}$ be given by $f(\mathbf{x}) := \prod_{i=1}^n x_i.$
\begin{itemize}
\item[(i)] Let $\psi : \mathbb{R}_{\geq 0} \to \mathbb{R}_{>0}$ be  regular and nonincreasing, and let $R$ be any element of $\mathbb{R}_{\geq 1}$ for which $\displaystyle \psi(R) < R^{\omega n}.$ Let $\eta$ denote the maximum norm on $\mathbb{R}^n.$ Then there exists some $C = C(\nu, \eta) \in \mathbb{R}_{\geq 1}$ such that for any real numbers $S$ and $T$ with $C  R \leq S \leq T,$ we have 
\begin{equation*}
m\left(\left\lbrace \mathbf{x} \in \mathbb{R}^n : 
f(\mathbf{x}) \leq \psi\big(\nu(\mathbf{x})\big) \ \text{and} \ S \leq \nu(\mathbf{x}) \leq T \right\rbrace\right) \asymp_{n, \omega, \psi, \nu} \,  \int_{S}^{T} \psi(t)^{1/\omega} \, t^{-1} \, {\log^{n-2}\left(t\right)} \, \mathrm{d} t. \end{equation*}
\item[(ii)] Let $\xi \in \mathbb{R}_{> 0}.$ Then there exists $N = N\left(\xi, n, \omega, \nu \right) \in \mathbb{Z}_{\geq 1}$ such that for any real numbers $S$ and $T$ with $N \leq S \leq T$ and any $\varepsilon \in \left(0, \frac{\xi}{2}\right) \subset \mathbb{R},$ we have 
\begin{equation*}
m\left(\left\lbrace \mathbf{x} \in \mathbb{R}^n : 
\left|f(\mathbf{x}) - \xi\right| \leq \varepsilon \ \text{and} \ S \leq \nu(\mathbf{x}) \leq T \right\rbrace\right) \asymp_{\xi, n, \omega, \nu} \,  \varepsilon \int_{S}^{T} t^{-1} \, {\log^{n-2}\left(t\right)} \, \mathrm{d} t.  \end{equation*}  
\end{itemize}
\end{lem}


\begin{proof} 
\begin{itemize}
\item[(i)]  
It follows from \cite[Corollary 4.2]{KS} and its proof that for any real numbers $S$ and $T$ with $R \leq S \leq T$, we have 
\begin{align*}
m\left(A_{f, \psi, \eta} \cap \{\mathbf{x} \in \mathbb{R}^n : {S} \leq\eta(\mathbf{x}) \leq {T} \} \right) &= m\left(A_{f^{1/\omega}, \psi^{1/\omega}, \eta} \cap \{\mathbf{x} \in \mathbb{R}^n : {S} \leq\eta(\mathbf{x}) \leq {T} \} \right) \\
&= 2^n \, n \int_{S}^{T} \frac{\psi({t})^{1/\omega}}{{t}}
 \left[ \sum_{i=0}^{n-2} \frac{1}{i!} \log^i\left(\frac{{t}^n}{\psi({t})^{1/\omega}}\right)\right] \mathrm{d} t \\
&\asymp_n \, \int_{S}^{T} \frac{\psi({t})^{1/\omega}}{{t}} \log^{n-2}\left(\frac{{t}^n}{\psi({t})^{1/\omega}}\right) \mathrm{d} t.    \end{align*} Since $\psi$ is regular and nonincreasing, there exist elements $\lambda_0 = \lambda_0(\psi)$ and $\lambda = \lambda(\psi)$ of $\mathbb{R}_{\geq 1}$ such that for every $t \in \mathbb{R}_{\geq 1},$ we have 
\[ \left(\lambda_0 t\right)^{-\lambda} < \psi(t) < \left(\lambda_0 t\right)^\lambda. \] Since $\log : \left( \mathbb{R}_{>0}, \cdot \right) \to \left( \mathbb{R}, + \right)$ is a group homomorphism, it follows that for any real numbers $S$ and $T$ with $R \leq S \leq T,$ we have \[ \int_{S}^{T} \frac{\psi({t})^{1/\omega}}{{t}} \log^{n-2}\left(\frac{{t}^n}{\psi({t})^{1/\omega}}\right) \mathrm{d} t  \asymp_{n, \omega, \psi} \, \int_{S}^{T} \psi(t)^{1/\omega} \, t^{-1} \, {\log^{n-2}\left(t\right)} \, \mathrm{d} t. \] Using the regularity of $\psi$ and the equivalence of all norms on $\mathbb{R}^n$ and arguing as in Theorem \ref{submerapprox}, the desired result follows.  

\item[(ii)] It follows from (i) that there exists $N = N\left(\xi, n, \omega, \nu \right) \in \mathbb{Z}_{\geq 1}$ such that for any real numbers $S$ and $T$ with $N \leq S \leq T$ and any $\theta \in \left[ \frac{\xi}{2}, \frac{3\xi}{2}  \right] \subset \mathbb{R}$, we have \begin{equation}\label{firstthetaxi}
m\left(\left\lbrace \mathbf{x} \in \mathbb{R}^n : f(\mathbf{x}) \leq \theta \ \text{and} \ S \leq \nu(\mathbf{x}) \leq T \right\rbrace\right) \asymp_{\xi, n, \omega, \nu} \,  \theta^{1/\omega} \int_{S}^{T} t^{-1} \, {\log^{n-2}\left(t\right)} \, \mathrm{d} t. 
\end{equation} 


\noindent Now let $\varepsilon \in \left(0, \frac{\xi}{2}\right) \subset \mathbb{R}$ be given. Applying \eqref{firstthetaxi} with $\theta = \xi + \varepsilon$ and $\theta = \xi - \varepsilon$ and then subtracting the latter resulting quantity from the former one implies that for any $S$ and $T$ as above, we have \begin{equation*}
m\left(\left\lbrace \mathbf{x} \in \mathbb{R}^n : \left|f(\mathbf{x}) - \xi\right| \leq \varepsilon \ \text{and} \ S \leq \nu(\mathbf{x}) \leq T \right\rbrace\right) \asymp_{\xi, n, \omega, \nu} \, \left[ \left(\xi + \varepsilon\right)^{1/\omega} - \left(\xi - \varepsilon\right)^{1/\omega} \right] \int_{S}^{T} t^{-1} \, {\log^{n-2}\left(t\right)} \, \mathrm{d} t.  \end{equation*} Since $\displaystyle \left(\xi + \varepsilon\right)^{1/\omega} - \left(\xi - \varepsilon\right)^{1/\omega} \asymp_{\xi, \omega} \, \varepsilon$, the desired result  follows. 
\end{itemize}
\end{proof}


\begin{lem}\label{subtractinequality}
Let $r$, $s$, $y$, $z$ be any elements of $\mathbb{R}$ that satisfy $0 < r < s$ and $1 \leq y < z.$ Then 
\begin{equation}\label{firstform}
    z^r - y^r < z^s - y^s. 
\end{equation}
\end{lem}
\begin{proof}
The inequality \eqref{firstform} is equivalent to the inequality \begin{equation}
    y^s - y^r < z^s - z^r,
\end{equation} which we now establish. Define $f : \mathbb{R}_{>0} \to \mathbb{R}$ by $\displaystyle f(t) := t^s - t^r.$ Set $\displaystyle c := \left(\frac{r}{s}\right)^{\frac{1}{(s-r)}}.$ Then $0 < c < 1.$ It is easy to check that for each $t \in (c, \infty) \subset \mathbb{R},$ we have $f'(t) > 0.$ This completes the proof. 
\end{proof}   





\item[(ii)] 
Suppose that $\psi$ is regular and nonincreasing and that $\displaystyle \lim_{t \to \infty} \psi(t) = 0.$ Then the pair $\left({_{ \xi} f}, \psi\right)$ is asymptotically acceptable. Fix $T_0 \in \mathbb{R}_{>0}$ for which $\psi\left(T_0\right) < \xi.$ Then \begin{equation}
m\left( A_{{_{ \xi} f}, \psi, \eta} \right) < \infty \quad \quad \text{if and only if} \quad \quad  \int_{T_0}^{\infty} \left[ \left(\xi + \psi(t)\right)^a - \left(\xi - \psi(t)\right)^a \right] {t}^{n - (r+1)} \ \mathrm{d} t < \infty. \end{equation} Moreover, if $a \geq 1,$ then \begin{equation}
m\left( A_{{_{ \xi} f}, \psi, \eta} \right) < \infty \quad \quad \text{if and only if} \quad \quad  \int_{T_0}^{\infty} \psi(t) \, {t}^{n - (r+1)} \ \mathrm{d} t < \infty. \end{equation}
\end{itemize}
\end{thm}







\begin{lem}\label{lemmakhintchine} Let $r \in \{1, \dots , n - 1 \}$ and $\mathbf{a} = (a_1, \dots , a_r) \in \left(\mathbb{R}_{>0}\right)^r$ be given. Define $f: \mathbb{R}^n \to \mathbb{R}$ by \[f(x_1, \dots , x_n) := \max\left\lbrace |x_i|^{a_i}: 1 \leq i \leq r\right\rbrace.\] Set $\displaystyle a := \sum_{i=1}^r  a_i^{-1}.$ Let $\xi \in \mathbb{R}_{> 0}.$ Let $\nu$ denote the maximum norm on $\mathbb{R}^n.$ Then there exists some $M = M(\xi, \mathbf{a}) \in \mathbb{R}_{>0}$ such that for any real numbers $S$ and $T$ with $M \leq S \leq T$ and any $\varepsilon \in \left(0, \frac{\xi}{2}\right) \subset \mathbb{R},$ we have 
\begin{equation*}\label{lemmaresult}
m\left(\left\lbrace \mathbf{x} \in \mathbb{R}^n : \left|f(\mathbf{x}) - \xi\right| \leq \varepsilon \ \text{and} \ S \leq \nu(\mathbf{x}) \leq T \right\rbrace\right) \asymp_{n, r, \xi, a} \, \varepsilon \int_{S}^{T} t^{n-(r+1)} \, \mathrm{d} t. 
\end{equation*}
\end{lem}
\begin{proof}
Let $\psi : \mathbb{R}_{\geq 0} \to \mathbb{R}_{>0}$ be any regular and nonincreasing function. Let $M_1 = M_1(\psi, \mathbf{a}) \in \mathbb{R}_{>0}$ be any element of $\mathbb{R}$ such that for each $t \in \mathbb{R}_{\geq M_1}$ and each $i \in \{1, \dots , r\},$ we have $\left(\psi(t)\right)^{1/a_i} < M_1.$ It then follows from \cite[Corollary 4.3 (i)]{KS} and its proof that for any real numbers $S_1$ and $T_1$ with $M_1 \leq S_1 \leq T_1,$ we have  \[ m\left(\left\lbrace \mathbf{x} \in \mathbb{R}^n : \left|f(\mathbf{x})\right| \leq \psi(\nu(\mathbf{x})) \ \text{and} \ S_1 \leq \nu(\mathbf{x}) \leq T_1 \right\rbrace\right) = 2^n \, (n-r) \int_{S_1}^{T_1} \left(\psi(t)\right)^a {t}^{n-(r+1)} \, \mathrm{d} t. \] It thus follows that there exists some $M = M(\xi, \mathbf{a})$ such that for any real numbers $S$ and $T$ with $M \leq S \leq T$ and any $\theta \in \left[ \frac{\xi}{2}, \frac{3\xi}{2}  \right] \subset \mathbb{R}$, we have 
\begin{equation}\label{thetaxi}
     m\left(\left\lbrace \mathbf{x} \in \mathbb{R}^n : \left|f(\mathbf{x})\right| \leq \theta \ \text{and} \ S \leq \nu(\mathbf{x}) \leq T \right\rbrace\right) = 2^n \, (n-r) \, \theta^a  \int_{S}^{T}  {t}^{n-(r+1)} \, \mathrm{d} t.
\end{equation}

\noindent Now let $\varepsilon \in \left(0, \frac{\xi}{2}\right) \subset \mathbb{R}$ be given. Applying \eqref{thetaxi} with $\theta = \xi + \varepsilon$ and $\theta = \xi - \varepsilon$ and then subtracting the latter resulting quantity from the former  yields \begin{equation*}
m\left(\left\lbrace \mathbf{x} \in \mathbb{R}^n : \left|f(\mathbf{x}) - \xi\right| \leq \varepsilon \ \text{and} \ S \leq \nu(\mathbf{x}) \leq T \right\rbrace\right) \asymp_{n, r} \, \left[ \left(\xi + \varepsilon\right)^a - \left(\xi - \varepsilon\right)^a \right] \int_{S}^{T} t^{n-(r+1)} \, \mathrm{d} t, 
\end{equation*} where $S$ and $T$ are in \eqref{thetaxi}. Since $\displaystyle \left(\xi + \varepsilon\right)^a - \left(\xi - \varepsilon\right)^a \asymp_{\xi, a} \varepsilon$, the result \eqref{lemmaresult} follows. 
\end{proof}



\begin{proof} 
Set $Z := h^{-1}\left(\mathbf{0}_{\mathbb{R}^\ell}\right) \neq \varnothing.$ For every $z \in Z,$ let $C_z \in \mathbb{R}_{>1},$ $V_z \subseteq \mathbb{R}^\ell$ with $\mathbf{0}_{\mathbb{R}^\ell} \in V_z,$ and $W_z \subseteq M$ with $z \in W_z$ be as in Lemma \ref{local}. Since $Z$ is compact, there exist finitely many $z_1, \dots , z_N \in Z$ such that $\displaystyle Z \subseteq \bigcup_{i=1}^N W_{z_i}. $ Set $\displaystyle C := \sum_{i=1}^N C_{z_i}.$ For each $i \in \{1, \dots , N \},$ let $Y_i$ be an open subset of $M$ with $z_i \in Y_i$ and for which $h$ is a submersion at each point of $Y_i.$ For each $i \in \{1, \dots , N \},$ set $U_i := Y_i \cap W_{z_i}.$ Since every submersion is an open map, it follows that $\displaystyle V := \bigcap_{i=1}^N \left( h\left(U_i\right) \cap V_{z_i} \right)$ is an open subset of $\mathbb{R}^\ell$ with $\mathbf{0}_{\mathbb{R}^\ell} \in V.$     


If $A$ is any Borel subset of $\mathbb{R}^\ell$ with $A \subseteq V,$ then \[ C^{-1} \, m(A) \le \min \big\{\sigma\big(W_{z_i}\cap h^{-1}(A)\big): 1\le i \le N\big\} \le \sigma\big(  h^{-1}(A)\big)\le  \sum_{i=1}^N \sigma\big(W_{z_i}\cap h^{-1}(A)\big)\le C \, m(A). \]  \qedhere  \end{proof}