\section{Introduction}
\subsection{Statements and Results}
Let $K$ be a number field and $S$ a finite set of places of $K$. Given two points $\alpha,\beta: \operatorname{Spec} \overline{K} \to \mathbb{P}^1_K$, we say that they are $S$-integral relative to each other if their Zariski closures in $\mathbb{P}^1_{O_K}$ do not intersect outside $S$. Let $\mu_{\infty}$ denote the set of all roots of unity $\zeta$, where $\zeta^n = 1$ for some $n$. In \cite{BIR08}, Baker, Ih and Rumely prove the following result.

\begin{theorem}[Theorem 2.1, \cite{BIR08}] \label{IntroBIR1}
Let $K$ be a number field and $S$ a finite set of places. For each $\beta \in \overline{K}^{\times} \setminus \mu_{\infty}$, the set of $\zeta \in \mu_{\infty}$ such that $\zeta$ is $S$-integral relative to $\beta$ is finite.
\end{theorem}

As observed in \cite{BIR08}, it is necessary that $\beta$ not be a root of unity. Our first result is a uniform version of Theorem \ref{IntroBIR1}. For an algebraic number $x \in \overline{\mathbb{Q}}$ and a number field $K$, we let $\deg_K(x) = |\operatorname{Gal}(\overline{K}/K) \cdot x|$ denote the size of the $\operatorname{Gal}(\overline{K}/K)$-orbit of $x$. 

\begin{theorem} \label{IntroUniformIh1}
Let $S$ be a finite set of places of $\mathbb{Q}$ and $K$ a number field. Then there exists a constant $C = C([K:\mathbb{Q}],S)$ such that the following holds: for any $\beta \in K^{\times} \setminus \mu_{\infty}$, if $\zeta \in \mu_{\infty}$ is $S$-integral relative to $\beta$ then $\deg_{\mathbb{Q}}(\zeta) < C$.
\end{theorem}

Thus for a fixed number field $K$, there are only finitely many roots of unity $\zeta$ that can be $S$-integral relative to any $\beta \in K^* \setminus \mu_{\infty}$. In \cite{GI13}, Grant and Ih show how one can deduce using results of Schinzel a uniform bound on the order of a root of unity $\zeta$ that is $S$-integral relative to a $S$-unit $\beta \in K^* \setminus \mu_{\infty}$. Here, our results hold without any condition on $|\beta|_v$. 
\par 
A natural question to ask is how $C$ depends on $[K:\mathbb{Q}]$. In \cite{Sil95}, a construction of Boyd is given which produces a sequence $\alpha_n \in \overline{\mathbb{Q}}$ such that $\alpha_n$ is $S$-integral relative to some root of unity of order $> \deg_{\mathbb{Q}}(\alpha_n)^{O(1/\log \log \deg_{\mathbb{Q}}(\alpha_n))}$, with $\deg_{\mathbb{Q}}(\alpha_n)$ tending to infinity. Hence one might expect that $C$ should grow polynomially with $[K:\mathbb{Q}]$. Our proof of Theorem \ref{IntroUniformIh1} only provides an upper bound that grows exponentially with $[K:\mathbb{Q}]$, but if one allows a small set of exceptions for each $\beta$, it is possible to get a polynomial bound.

\begin{theorem} \label{IntroUniformIh3}
Let $S$ be a finite set of places of $\mathbb{Q}$. Then there exists a constant $c > 0$, such that for all $\beta \in \overline{\mathbb{Q}}^{\times} \setminus \mu_{\infty}$, the set
$$\{\zeta \in \mu_{\infty} : \deg_{\mathbb{Q}}(\zeta) > c \deg_{\mathbb{Q}}(\beta)^{10} \text{ and } \zeta \text{ is S-integral relative to } \beta\}$$
is a union of at most $|S_{\operatorname{fin}}|$ $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-orbits.
\end{theorem}

For example, if $S = \{2,3,\infty\}$, then for each $\beta$ with $[\mathbb{Q}(\beta):\mathbb{Q}] \leq D$, except for two possible $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-orbits as exceptions, all other roots of unity $\zeta$ that are $S$-integral relative to $\beta$ must have $< cD^{10}$ conjugates, where $c > 0$ is a constant independent of $D$ and $\beta$. 
\par 
Baker--Ih--Rumely also prove an analogous result for elliptic curves $E$, where $\mu_{\infty}$ is replaced by the set of all torsion points of $E$. They then conjecture a dynamical analogue of their result. For a rational map $\varphi: \mathbb{P}^1 \to \mathbb{P}^1$ defined over a field $K$, we say that a point $x \in \mathbb{P}^1(\overline{K})$ is preperiodic if there exist distinct positive integers $m,n$ such that $\varphi^m(x) = \varphi^n(x)$. We will denote the set of preperiodic points of $\varphi$ by $\operatorname{Prep}(\varphi)$. 

\begin{conjecture} \label{Ih1} (Ih's Conjecture) 
Let $\varphi: \mathbb{P}^1 \to \mathbb{P}^1$ be a rational map of degree $d \geq 2$ defined over a number field $K$ and $S$ a finite set of places of $K$. For any non-preperiodic point $\beta \in \mathbb{P}^1(K)$, there are only finitely many preperiodic points $x \in \mathbb{P}^1(\overline{K})$ that are $S$-integral relative to $\beta$.
\end{conjecture}

Ih's conjecture has been proven for power and Lattès maps by Baker--Ih--Rumely \cite{BIR08}, and for Chebyshev Maps by Ih-Tucker \cite{IT10}. Petsche \cite{Pet08} has established the conjecture for any rational map $\varphi$ of degree $d \geq 2$, but under the additional assumption that $\beta$ is a totally Fatou point. Quantitative results in Petsche's setting have recently been obtained by Young \cite{You22}. Our next two theorems are analogues of Theorems \ref{IntroUniformIh1} and \ref{IntroUniformIh3} for Lattès maps. Here, a Lattès map $\varphi: \mathbb{P}^1 \to \mathbb{P}^1$ is a rational map such that we have the following commutative diagram:
$$
\begin{tikzcd}
E \arrow{r}{\psi} \arrow{d}[swap]{\pi}  & E \arrow{d}[swap]{\pi}  \\
\mathbb{P}^1 \arrow{r}{\varphi} & \mathbb{P}^1
\end{tikzcd}
$$
where $E$ is an elliptic curve, $\psi: E \to E$ is a self map (not necessarily a group homomorphism) with degree $d \geq 2$ and $\pi: E \to \mathbb{P}^1$ a non-constant map. Our next result is the analogue of Theorem \ref{IntroUniformIh1}, but only for Lattès maps coming from elliptic curves with complex multiplication.

\begin{theorem} \label{IntroUniformIh2}
Let $K$ be a number field and $S$ be a finite set of places of $K$. Let $\varphi: \mathbb{P}^1 \to \mathbb{P}^1$ be a Lattès map defined over $K$ that is associated to an elliptic curve $E$ with complex multiplication and let $L$ be a finite extension. Then there exists a constant $C = C([L:K],S)$ such that the following holds: for any $\beta \in \mathbb{P}^1(L) \setminus \operatorname{Prep}(\varphi)$, if $x \in \operatorname{Prep}(\varphi)$ is $S$-integral relative to $\beta$ then $\deg_{K}(x) < C$.
\end{theorem}

The assumption that our elliptic curve has complex multiplication is needed in our proof of Theorem \ref{IntroUniformIh2}. For our analogue of Theorem \ref{IntroUniformIh3} where we allow finitely many exceptions, we are able to prove a version that holds for all Lattès maps without any assumption of complex multiplication.

\begin{theorem} \label{IntroUniformIh4}
Let $K$ be a number field and $S$ be a finite set of places of $K$. Let $\varphi: \mathbb{P}^1 \to \mathbb{P}^1$ be a Lattès map defined over $K$. Then there exists a constant $c > 0$, such that for all non-preperiodic $\beta \in \mathbb{P}^1(\overline{K})$, the set
$$\{x \in \operatorname{Prep}(\varphi) : \deg_{K}(x) > c \deg_{K}(\beta)^{20} \text{ and } x \text{ is S-integral relative to } \beta\}$$
is a union of at most $|S|$ $\operatorname{Gal}(\overline{K}/K)$-orbits.
\end{theorem}





\subsection{Overview of Proofs} 
The main idea is the notion of logarithmic equidistribution. Let $\varphi: \mathbb{P}^1 \to \mathbb{P}^1$ be a rational map defined over our number field $K$ and for each place $v \in M_K$, let $\mu_{\varphi,v}$ be the equilibrium measure of $\varphi$ for $v$. For archimedean $v$, this is a probability measure supported on the Julia set of $\varphi$ on $\mathbb{P}^1(\mathbb{C})$, and in general it is a probability measure on the Berkovich analyitificaion $\mathbb{P}^1_{\operatorname{Berk},v}$ (see \cite{BR10} for more details). 
\par 
Let $h(x)$ denote the usual logarithmic Weil height on $\mathbb{P}^1(\overline{\mathbb{Q}})$ and let
$$h_{\varphi}(x) = \lim_{n \to \infty} \frac{1}{d^n} h(\varphi^n(x))$$
be the canonical height for $\varphi$ where $d$ is the degree of $\varphi$. By work of Baker--Rumely \cite{BR06}, Chambert-Loir \cite{CL06} and Favre--Rivera-Letelier \cite{FRL06}, it is known that for a sequence of distinct points $(x_n) \in \mathbb{P}^1(\overline{K})$ with dynamical height $h_{\varphi}(x_n)$ tending to zero, the Galois orbits of $x_n$ equidistribute to our equilibrium measure $\mu_{\varphi,v}$. More precisely, if one lets $F_n$ be the Galois orbit of $x_n$ over $K$, then for any continuous function $f$ on $\mathbb{P}^1_{\operatorname{Berk},v}$, we have 
$$\lim_{n \to \infty} \frac{1}{|F_n|} \sum_{x \in F_n} f(x) \to \int f d\mu_{\varphi,v}.$$
\par 
Let $\beta \in \mathbb{P}^1(K)$. If one has the above convergence also for the function $f(x) = \log^+|x-\beta|^{-1}_v$, we say that the sequence $(F_n)$ satisfies logarithmic equidistribution at $\beta$. One can then show that if $(F_n)$ satisfies logarithmic equidistribution at $\beta$, then only finitely many $x_n$'s can be $S$-integral relative to $\beta$. This is the strategy used by Baker--Ih--Rumely \cite{BIR08} and by Petsche \cite{Pet08}. Outside the case of power and Lattès maps, logarithmic equidistribution has also been established by Szpiro--Tucker \cite{ST11} in the setting of taking $F_n = \varphi^{-n}(\alpha)$ as a whole, instead of looking at individual Galois orbits. This has also been used in other contexts, such as by Benedetto--Ih in studying $S$-integral points on a moduli space \cite{BI20} or by Dimitrov--Habbeger on $S$-integral points on $\mathbb{G}_m^n$ \cite{DH19}.
\par 
To obtain uniform results, we will have to bound the rate of logarithmic equidistribution in terms of the Weil height $h(\beta)$. We follow the approach of Baker--Ih--Rumely, where we break into two steps. The first step is to bound how close a preperiodic point $x$ can be to $\beta$, in terms of the size of the Galois orbit of $x$. The second is to bound how many points in the Galois orbit of $x$ can be close to $\beta$. In \cite{BIR08}, the second step is done by an equidistribution theorem that is specific to the cases of $\varphi(z) = z^d$ and elliptic curves. We however use a general quantitative equidistribution theorem due to Favre and Rivera-Letelier \cite{FRL06} and are able to do the second step for any rational map $\varphi(z)$ of degree $d \geq 2$ (see Lemma \ref{GeneralQuantLog2} for a precise statement). This reduces our problem to showing the existence of a constant $C > 0$ along with a fixed $\delta > 0$ such that for any Galois orbit $F$ of preperiodic points, we have
$$\max_{x \in F} \log^+|x-\beta|_v^{-1} < C(h(\beta)+1)|F|^{1/2 - \delta}.$$
When $\varphi(z) = z^d$, such a bound can be obtained using linear forms in logarithms. We then use a result due to Laurent, Mignotte and Nesterenko \cite{LMN95} which give us the bound we want. 
\par 
In the case of Lattès maps, there is a theory of linear forms in elliptic logarithms. Using results of David--Hirata-Kohno \cite{SH09}, it turns out that in general, the bounds obtained are not strong enough to prove a version of Theorem \ref{IntroUniformIh1} for Lattès maps as there is an extra factor of a power of $\log^+ h(\beta)$ that appears in our upper bound on $\log^+|x-\beta|_v^{-1}$. This forces us to allow the possible existence of an exceptional Galois orbit $F$ for each place that could be $S$-integral relative to $\beta$ in order to strengthen our upper bound. For elliptic curves with complex multiplication, Ably and Gaudron \cite{AG03} have proven stronger bounds where there is no extra $\log^+ h(\beta)$ factor, which allows us to obtain Theorem \ref{IntroUniformIh2} for CM elliptic curves.


\subsection{Acknowledgements} The author would like to thank Laura DeMarco, Niki Myrto Mavraki and Yan Sheng Ang for helpful discussions about the paper and the problem.






\section{Quantitative Equidistribution}
We now recall the quantitative version of equidistribution as proven by Favre--Rivera-Letelier in \cite{FRL06}. Such quantitative equidistribution methods have been used in \cite{Fil17}, \cite{DKY20}, \cite{DKY21} and \cite{You22} to obtain uniform results. We first introduce the Berkovich projective line $\mathbb{P}^1_{\operatorname{Berk},v}$.


\subsection{The Berkovich Projective Line $\mathbb{P}^1_{\operatorname{Berk},v}$} Let $\mathbb{C}_v$ be an algebraically closed field that is complete with respect to a non-archimedean valuation $v$. When attempting to study dynamics of a rational map $\varphi: \mathbb{P}^1(\mathbb{C}_v) \to \mathbb{P}^1(\mathbb{C}_v)$, it turns out that it is more natural to study the situation over the Berkovich analytification $\varphi: \mathbb{P}^1_{\operatorname{Berk},v} \to \mathbb{P}^1_{\operatorname{Berk},v}$. Interested readers may consult \cite{Ben19} for a thorough treatment of the theory. 
\par 
Let $D(a,r) \subseteq \mathbb{A}^1(\mathbb{C}_v)$ be the open disc that is centered at $a$ with radius $r$. Each such disc corresponds to a point on $\mathbb{P}^1_{\operatorname{Berk},v}$. When $r = 0$, we can identify this with the usual point $a \in \mathbb{P}^1(\mathbb{C}_v)$. This are known as the Type I points, or also as the classical points. Let $\zeta(a,r)$ denote the point on $\mathbb{P}^1_{\operatorname{Berk},v}$ corresponding to the disc $D(a,r)$. When $r$ is an element of the value group $|\mathbb{C}_v|^{\times}$, we say that $\zeta(a,r)$ is a Type II point. If it is not, we say that $\zeta(a,r)$ is a Type III point. The remaining points of $\mathbb{A}^1_{\operatorname{Berk},v}$ are called Type IV points, and they correspond to a nested intersection of discs $\cdots \subseteq D_n \subseteq \cdots \subseteq D_1$ such that $\cap_{n=1}^{\infty} D_n = \emptyset$ but their radii do not go to zero. Finally, $\mathbb{P}^1_{\operatorname{Berk},v}$ consists of $\mathbb{A}^1_{\operatorname{Berk},v}$ along with an extra type I point called $\infty$. 
\par 
Any element of $f(z) \in \mathbb{C}_v[z]$ extends to a function on $\mathbb{A}^1_{\operatorname{Berk},v} \to \mathbb{R}_{\geq 0}$ and the topology on $\mathbb{A}^1_{\operatorname{Berk},v}$ is given the weakest topology such that all polynomials $f(z)$ are continuous on $\mathbb{A}^1_{\operatorname{Berk},v}$. This makes $\mathbb{P}^1_{\operatorname{Berk},v}$ into a compact and path-connected space where $\mathbb{P}^1(\mathbb{C}_v)$ sits inside as a dense subset. 
\par 
The Berkovich space allows one to develop a suitable analogue of the Laplacian $\triangle = \frac{1}{2\pi} (\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2})$ for non-archimedean places. This is developed in full in Baker-Rumely \cite{BR10}. For a suitable class of continuous functions $f: \mathbb{P}^1_{\operatorname{Berk},v} \to \mathbb{R}$, its Laplacian $\triangle f$ is a signed Borel measure on $\mathbb{P}^1_{\operatorname{Berk},v}$ of total mass zero. Also, it is a self-adjoint operator in the sense that
$$\int f \triangle g = \int g \triangle f.$$
This fact will be repeatedly used later in computations. Here, we note that our Laplacian is normalized so that $\triangle \log |z| = \delta_{0} - \delta_{\infty}$. Using non-archimedean potential theory, it is then possible to construct an analogue of the equilibrium measure $\mu_{\varphi,v}$ for a given rational map $\varphi: \mathbb{P}^1_{\operatorname{Berk},v} \to \mathbb{P}^1_{\operatorname{Berk},v}$ of degree $d \geq 2$. For example, if $\varphi$ has good reduction, it is simply the delta mass at the Gauss point $\zeta(0,1)$.


\subsection{Quantitative Equidistribution following Favre--Rivera-Letelier }
Let $K$ be a number field, $K_v$ its completion for a place $v$ and $\mathbb{C}_v$ the completion of $\overline{K}_v$. Let $M_K$ denote the places of $v$. We first introduce some definitions from \cite{FRL06}. For each $v \in M_K$, let $\rho_v$ be a measure on $\mathbb{P}^1_{\operatorname{Berk},v}$.
\par 
We say that $\rho_v$ has a continuous potential if $\rho_v = \lambda_v + \triangle g$ for some continuous function $g$ which is the difference of two continuous subharmonic functions. Here, $\lambda_v$ is the Gauss point $\zeta(0,1)$ for non-archimedean $v$ and is the uniform probability measure supported on the unit circle for archimedean $v$. We say that $\rho_v$ has a Hölder-continuous potential with exponent $\kappa$ with respect to a metric $d$ if furthermore there exists a constant $C > 0$ such that
$$|g(z)-g(w)| \leq Cd(z,w)^{\kappa}$$
for all classical points $z,w \in \mathbb{P}^1(\mathbb{C}_v)$. 
\par 
A collection of measures $\rho = (\rho_v)_{v \in M_K}$ is said to be an adelic measure if each $\rho_v$ has a continuous potential and $\rho_v = \lambda_v$ for all but finitely many $v$. Given two measures $\rho_v,\rho_v'$ on $\mathbb{P}^1_{\operatorname{Berk},v}$, we define a bilinear form by
$$( \rho_v, \rho_v')_v = -\int_{\mathbb{A}^1_{\operatorname{Berk},v} \times \mathbb{A}^1_{\operatorname{Berk},v} \setminus \operatorname{Diag}} \log|z-w|_v d \rho_v(z) d \rho'_v(w)$$
where $\operatorname{Diag} = \{(z,z), z \in \mathbb{C}_v\}$ is the diagonal of classical points. This integral exists if both $\rho_v,\rho_v'$ either have a continuous potential or are discrete point masses. For a finite set of points $F \subset \overline{K}$ that is $\operatorname{Gal}(\overline{K}/K)$-invariant, we can then define its height with respect to $\rho$ as
$$h_\rho(F) = \frac{1}{2} \sum_{v \in M_K} (\!( [F]-\rho_v,[F]-\rho_v )\!)_v$$
where $[F] = \frac{1}{|F|} \sum_{x \in F} \delta_{x}$ and $(\!(\, \, , \, \,)\!) = N_v(\, \, , \, \,)$ with $N_v = [K_v:\mathbb{Q}_v]/[K:\mathbb{Q}]$. 
\par 
For an infinite place $v$, we say a continuous function $f: \mathbb{P}^1(\mathbb{C}_v)\to \mathbb{R}$ is of class $C^k_{\operatorname{sph}}$ if it is $C^k$ with respect to the spherical metric 
$$d_{\operatorname{sph}}(x,y)= \frac{|x_1 y_2 - x_2 y_1|_v}{\max\{|x_1|_v, |x_2|_v\} \max\{ |y_1|_v, |y_2|_v\}}$$
where $x = [x_1:x_2]$ and $y = [y_1:y_2]$. For a finite place $v$, we say $f: \mathbb{P}^1_{\operatorname{Berk},v} \to \mathbb{R}$ is of class $C^k_{\operatorname{sph}}$ if it is locally constant outside of a finite subtree $T \subset \mathbb{H}_v$ and $T$ is a finite union of segments where $f$ is of the usual class $C^k$ on. Given $f$ of class $C^k_{\operatorname{sph}}$ for $k \geq 1$, we define 
$$\langle f , f \rangle_v =  \int_{\mathbb{C}}\left(\frac{\partial f}{\partial x} \right)^2 + \left(\frac{\partial f}{\partial y} \right)^2 dx dy$$
if $v$ is archimedean. If $v$ is non-archimedean, we fix a basepoint $S_0 \in \mathbb{H}_v$ and let $\partial f(S)$ be the derivative of $f$ restricted to the segment $[S_0,S]$. Then we define
$$ \langle f , f \rangle_v = \int_{\mathbf{P}^1(\mathbb{C}_v)} (\partial f)^2 d \lambda.$$
We note that these definitions of energy still work if one assumes $f$ is merely continuous, if we use the weak derivative instead.
We can now state the quantitative equidistribution result of Favre--Rivera-Letelier. 

\begin{theorem}[Theorem 7, \cite{FRL06}]  \label{QuantEquib1} 
Let $\rho = \{\rho_v\}_{v \in M_K}$ be an adelic measure where each $\rho_v$  has a Hölder-continuous potential of exponent $\kappa \leq 1$ with respect to the spherical metric. Then there exists a constant $C > 0$, only depending on $\rho$, such that for all places $v$ and all functions $f$ of class $C^1_{\operatorname{sph}}$ on $\mathbb{P}^1_{\operatorname{Berk},v}$, and for all finite $\operatorname{Gal}(\overline{K}/K)$-invariant sets $F$, we have
$$\left| \frac{1}{|F|} \sum_{\alpha \in F} f(\alpha) - \int_{\mathbb{P}^1_{\operatorname{Berk},v}} f d\rho_v \right| \leq \frac{\operatorname{Lip}_{\operatorname{sph}}(f)}{|F|^{1/\kappa}} + \left(h_{\rho}(F) + C \frac{\log|F|}{|F|} \right)^{1/2} \langle f ,f \rangle_v^{1/2}.$$
Here, $\operatorname{Lip}_{\operatorname{sph}}$ is the Lipschitz constant for $f$ with respect to the spherical metric.
\end{theorem}

We wish to apply the theorem to functions that are Hölder-continuous and not necessarily differentiable. Thus we have to extend the theorem to a larger class of test functions. Also, we want to replace the spherical metric with the standard Euclidean metric, which we do so at the cost of assuming $\infty \not \in F$.

\begin{proposition} \label{QuantEquib2}
Let $\rho = \{\rho_v\}_{v \in M_K}$ be an adelic measure where each $\rho_v$ has a Hölder-continuous potential of exponent $\kappa \leq 1$. Then there exists a constant $C > 0$, only depending on $\rho$, such that for all places $v$ and all continuous functions $f$ on $\mathbb{P}^1_{\operatorname{Berk},v}$ such that $\triangle f$ has a Hölder continuous potentials of exponent $\kappa'$ with respect to the Euclidean metric $|x-y|_v$, and for all finite $\operatorname{Gal}(\overline{K}/K)$-invariant sets $F$ not containing $\infty$, we have
$$\left| \frac{1}{|F|} \sum_{\alpha \in F} f(\alpha) - \int_{\mathbb{P}^1_{\operatorname{Berk},v}} f d \rho_v \right| \leq \frac{\operatorname{Hol}_{\kappa'}(f)}{|F|^{\kappa'/\kappa}} + (h_{\rho}(F) + C \frac{\log |F|}{|F|} )^{1/2} (\triangle f, \triangle f)_v^{1/2}$$
where $\operatorname{Hol}_{\kappa'}(f)$ is a constant for the Hölder continuity of $f$ with exponent $\kappa'$.
\end{proposition}

\begin{proof}
We follow the proof of Theorem 7 of \cite{FRL06}. The first part of the proof still applies, giving us the inequality
$$(\!([F]_{\epsilon} - \rho_v, [F]_{\epsilon} - \rho_v)\!)_v \leq h_{\rho}(F) + M(2\eta(\epsilon) + |F|^{-1}(C  + \log \epsilon^{-1})$$
where $\eta(\epsilon) = \hat{\eta}(\epsilon) + \epsilon$, with $\hat{\eta}(\epsilon)$ being a modulus of continuity for all places $v$ where $\rho_v \not = \lambda_v$. Here as our points lie in $\mathbb{A}^1(\mathbb{C}_v)$, we can regularize using the standard metric $|x-y|_v$ and so $\hat{\eta}$ can be taken with respect to the Euclidean metric $|x-y|_v$. Then as $\eta(\epsilon) \leq C' \epsilon^{\kappa}$ for some $C' > 0$, taking $\epsilon = |F|^{-1/\kappa}$ gives us
$$(\!( [F]_{\epsilon} -\rho_v, [F]_{\epsilon}-\rho_v )\!)_v \leq h_{\rho}(F) + C\frac{\log|F|}{|F|}$$
for a new constant $C > 0$ that depends only on $\rho$. Now observe that 
$$\int f d([F]_{\epsilon} - \rho_v) = \int \int \log|x-y|_v \triangle f d([F]_{\epsilon}-\rho_v) = (\triangle f, [F]_{\epsilon} - \rho_v).$$
Thus by Cauchy-Schwarz, we have
$$\left|\int f d([F]_{\epsilon} - \rho_v) \right| \leq (\triangle f, \triangle f)^{1/2}([F]_{\epsilon} - \rho_v, [F]_{\epsilon} - \rho_v)^{1/2}$$
$$\leq (h_{\rho}(F) + C \frac{\log |F|}{|F|})^{1/2} (\triangle f, \triangle f)_v^{1/2}.$$
Now we just need to bound $\left|\int f d([F]_{\epsilon}-[F]) \right|$, which is bounded by $\operatorname{Hol}_{\kappa'}(f)\epsilon^{\kappa'}$. Since $\epsilon = |F|^{-1/\kappa}$, our proposition follows.
\end{proof}

Let $F$ be a $\operatorname{Gal}(\overline{K}/K)$-invariant set of preperiodic points. For a place $v$, we  want to estimate $|F| \cap D(x,\epsilon)$ where $D(x,\epsilon)$ is the disc of radius $\epsilon$ centered at $x$. By equidistribution, as $|F| \to \infty$ we should get $\mu(D(x,\epsilon))|F|$ many points, where $\mu$ is the equilibrium measure for $v$. We will use quantitative equidistribution to obtain an upper bound.

\begin{proposition} \label{QuantDisc1}
Let $F$ be a $\operatorname{Gal}(\overline{K}/K)$-invariant set of preperiodic points with $|F| = m$ and let $\kappa$ be a Hölder exponent for $\mu_{\varphi,v}$. Then there exists a constant $C > 0$ such that for any $\epsilon > 0$, we have 
$$|F \cap D(x,\epsilon)| \leq \frac{1}{\epsilon m^{1/\kappa - 1}} + m \cdot \mu_{\varphi,v}(D(x,e \cdot \epsilon)) + C \sqrt{m \log m}.$$
Here, we interpret $D(\beta, \epsilon)$ as an open Berkovich disc in the Berkovich space $\mathbb{P}^1_{\operatorname{Berk},v}$.
\end{proposition}

\begin{proof}
Without loss of generality we may assume that $x = 0$. For $v$ archimedean, consider the function
$$f(z) = \min(1, \max(0,  \log|\frac{z}{\epsilon}|))$$
and for $v$ non-archimedean, let $f(z)$ be the continuous function on $\mathbb{P}^1_{\operatorname{Berk},v}$ that linearly interpolates $0$ on $\zeta(0,\epsilon)$ and $1$ on $\zeta(0, e \epsilon)$, and whose value on all other points $x$ is the value at the retraction of $x$ to the path between $\zeta(0,\epsilon)$ and $\zeta(0, e \epsilon)$. Then $\triangle f = \delta_{\zeta(0, \epsilon)} - \delta_{\zeta(0, e \epsilon)}$ and $0 \leq f \leq 1$. One may then check that $(\triangle f , \triangle f) = \log e = 1$ for any place $v$.
\par 
Applying Proposition \ref{QuantEquib2}, for any Hölder exponent $\kappa \leq 1$, we get a constant $C > 0$ such that 
$$\left| \frac{1}{m} \sum_{\alpha \in F} f(\alpha) - \int f d\mu_{\varphi,v} \right| \leq \frac{\operatorname{Lip}_{\operatorname{std}}(f)}{m^{1/\kappa}} + (h_{\varphi}(F) + C \frac{\log m}{m})^{1/2} ( \triangle f, \triangle f )_v^{1/2}.$$
Since $F$ consists of preperiodic points, we have $h_{\varphi}(F) = 0$. The Lipschitz constant may be bounded by $\frac{1}{\epsilon}$ in both archimedean and non-archimedean cases, and we have $\langle f,f \rangle \leq 2$. Using the fact that $\int f d \mu_{\varphi,v} \leq \mu_{\varphi,v}(D(0,e \epsilon))$ by construction, we get
$$\frac{1}{m} \sum_{\alpha \in F_n} f(\alpha) \leq \frac{1}{\epsilon m^{1/\kappa}} + \mu_{\varphi,v}(D(0,e \epsilon)) + C \frac{\sqrt{\log m}}{\sqrt{m}}$$
where we absorbed $2$ into the constant $C$. Multiplying $m$ over, we have
$$\sum_{\alpha \in F_n} f(\alpha) \leq \frac{1}{\epsilon m^{1/\kappa - 1}} + m \cdot \mu_{\varphi,v}(D(0,e \epsilon)) + C \sqrt{m \log m}.$$
Finally we observe that for any $x \in D(0,\epsilon)$, we have $f(x) = 1$ and so it follows that 
$$|F \cap D(0,\epsilon)| \leq \frac{1}{\epsilon m^{1/\kappa - 1}} + m \cdot \mu_{\varphi,v}(D(0,e \epsilon)) + C \sqrt{m \log m}.$$

\end{proof}



\section{The Arakelov--Zhang Pairing and S-Integrality}
In this section, we introduce the Arakelov--Zhang pairing on $\mathbb{P}^1$, following the approach of \cite{PST12}. We then use it to relate logarithmic equidistribution with finiteness of $S$-integral points and also establish quantative bounds on the convergence to the pairing.

\subsection{The Arakelov--Zhang Pairing on $\mathbb{P}^1$} 
As always, let $K$ be a number field, $M_K$ the set of places of $K$ and $\mathbb{C}_v$ the completion of $\overline{\mathbb{Q}}_v$. 
\par 
Let $L$ be a line bundle on $\mathbb{P}^1$. For each place $v \in M_K$, the line bundle $L$ extends to a line bundle $L_v$ on $\mathbb{P}^1_{\operatorname{Berk},v}$. A continuous metric $|| \cdot ||_v$ on $L_v$ is a continuous function $|| \cdot ||_v: L_v \to \mathbb{R}_{\geq 0}$ that induces a norm on each fiber $L_x$ as a $\mathbb{C}_v$-vector space. The metric is said to be semi-positive if for a section $s(x)$, the function $\log |s(x)|: \mathbb{P}^1 \to \mathbb{R} \cup \{-\infty\}$ is subharmonic and is said to be integrable if $\log |s(x)|$ can be written as a difference of two subharmonic functions.
\par 
For $L = O(1)$, there is a standard metric $|| \cdot ||_{\operatorname{std},v}$ given by
$$||s(x)||_{\operatorname{std},v} = \frac{|s(x_0,x_1)|_v}{\max\{|x_0|_v,|x_1|_v\}}$$
where $s(x_0,x_1)$ is the linear polynomial in $\mathbb{C}_v[x_0,x_1]$ representing the section $s$ and $x = [x_0:x_1]$. For $L = O(1)$, an integrable adelic metric is a family of metrics where $|| \cdot ||_v$ is a continuous integrable metric for $L$ over $\mathbb{P}^1_{\operatorname{Berk},v}$. We further impose that for all but finitely many $v$, we have $|| \cdot ||_v = || \cdot ||_{\operatorname{std},v}$.
\par
Given a finite set $F \subset \mathbb{P}^1(\overline{K})$ which is $\operatorname{Gal}(\overline{K}/K)$-invariant and a section $s$ of $O(1)$ with $\operatorname{div}(s) \not \in F$, we can define its height with respect to our adelic line bundle $\overline{L}$ by 
$$h_{\overline{L}}(F) = \frac{1}{|F|} \sum_{x \in F} \sum_{v \in M_K} N_v\log ||s(x)||_{v}^{-1}$$
where $N_v = \frac{[K_v:\mathbb{Q}_v]}{[K:\mathbb{Q}]}$. For a rational map $\varphi: \mathbb{P}^1 \to \mathbb{P}^1$ defined over $K$ with degree $d \geq 2$, let $\overline{L}_{\varphi}$ be a canonical adelic line bundle for $\varphi$ (3.5, \cite{PST12}). The height $h_{\overline{L}_{\varphi}}$ agrees with the canonical height $h_{\varphi}$. 
\par 
Let $\overline{L}_1, \overline{L}_2$ be two semi-positive adelically metrized line bundles with $O(1)$ as its underlying bundle. Let $s,t$ be two sections with $\operatorname{div}(s) \cap \operatorname{div}(t) = \emptyset$. For a place $v$, we can define the local Arakelov-Zhang pairing between $\overline{L}_1$, $\overline{L}_2$ with respect to $s$ and $t$ as 
$$\langle \overline{L}_{1}, \overline{L}_{2} \rangle_{s,t,v} = \int_{\mathbb{P}^1_{\operatorname{Berk},v}} \log ||s(x)||_{v,1}  \triangle \log ||t(x)||_{v,2} $$
$$= \log||s(\operatorname{div}(t))||_{v,1} - \int_{\mathbb{P}^1_{\operatorname{Berk},v}} \log ||s(x)||_{v,1} d \mu_{\overline{L}_{v,2}}(x)$$
where $\triangle \log ||t(x)||_{v,2} = \operatorname{div}(t) - d\mu_{\overline{L}_{v,2}}$ and $d \mu_{\overline{L}_{v,2}}$ is independent of $t$. Note that we integrated over $\log |s(x)|$ instead of $\log |s(x)|^{-1}$ as our Laplacian differs from that of \cite{PST12} by a negative sign. The global Arakelov--Zhang pairing is then defined as
$$\langle \overline{L}_1, \overline{L}_2 \rangle_{s,t} = \sum_{v \in M_K} N_v \langle \overline{L}_{1}, \overline{L}_{2} \rangle_{s,t,v} + h_{\overline{L}_1}(\operatorname{div} (t)) + h_{\overline{L}_2}(\operatorname{div}(s))$$
$$= h_{\overline{L}_2}(\operatorname{div}(s)) + \sum_{v \in M_K} N_v \int_{\mathbb{P}^1_{\operatorname{Berk},v}} \log ||s(x)||_{v,1}^{-1} d \mu_{\overline{L}_{v,2}}(x).$$
Unlike the local pairing, this does not depend on the choice of $s$ and $t$ and we may remove it from the notation.

\begin{theorem}[Theorem 11, \cite{PST12}] \label{Pairing1}  Let $\varphi$ be a rational function of degree $d \geq 2$ on $\mathbb{P}^1(K)$ and let $\overline{L}_\varphi$ be a canonical adelization for $O(1)$. Then for any other adelically metrized $\overline{L}$, for any sequence of $\operatorname{Gal}(\overline{K}/K)$-invariant multiset $(F_n)$ with $h_{\overline{L}_\varphi}(F_n) \to 0$, we have
$$h_{\overline{L}}(F_n) \to \langle \overline{L}, \overline{L}_\varphi \rangle.$$

\end{theorem}

The proof of Theorem \ref{Pairing1} is exactly the same as Theorem 11 of \cite{PST12}. To be precise, the proof tells us that $$h_{\overline{L}}(F_n) \to \langle \overline{L}, \overline{L}_{\varphi} \rangle - \langle \overline{L}_{\varphi}, \overline{L}_{\varphi} \rangle$$
and the second term is zero by Proposition 10 of \cite{PST12}. We note that the proof tells us the rate of convergence is controlled by $h_{\overline{L}_{\varphi}}(F)$ along with the rate of equidistribution of the function $f_v = \frac{ \log ||  s(x)||_{v,\epsilon,\varphi}}{\log || s(x)||_{v}}$ where $\epsilon$ is some polarization for $\varphi$. This gives us the following corollary of the proof.

\begin{corollary} \label{Pairing2}
Let $\varphi, \overline{L}_{\varphi}, \overline{L}$ and $F_n$ be as in Theorem \ref{Pairing1}. Then

$$|h_{\overline{L}}(F_n) - \langle \overline{L}, \overline{L}_{\varphi} \rangle | \leq h_{\varphi}(F_n) + \sum_{v \in M_K} N_v \left| \frac{1}{|F|} \sum_{x \in F} f_v(x)  - \int f_v(x) d \mu_{\varphi}(x) \right|.$$
\end{corollary}

\subsection{Logarithmic Equidistribution and S-integrality} For $x,y \in \mathbb{P}^1(\mathbb{C}_v)$, let 
$$d_{\operatorname{sph}}(x,y) = \frac{|x_1y_2 - x_2 y_1|_v}{\max\{|x_1|_v, |x_2|_v \max\{|y_1|_v, |y_2|_v\}}$$
be the $v$-adic chordal metric where $x = [x_1:x_2]$ and $y = [y_1:y_2]$. The logarithmic chordal metric $\lambda_v(x,y)$ is defined to as 
$$\lambda_v(x,y) = -\log d_{\operatorname{sph}}(x,y).$$
The function $\lambda_v(x,y)$ is non-negative and it detects $S$-integrality in the following manner. Given two points $\alpha,\beta: \operatorname{Spec} \overline{K} \to \mathbb{P}^1_K$, the Zariski closure of $x$ and $y$ in $\mathbb{P}^1_{O_K}$ do not intersect above the place $v$ if and only if $\lambda_v(\alpha',\beta') = 0$ for all Galois conjugates $\alpha',\beta'$ of $\alpha,\beta$. 
\par 
Let $\beta \in K$ and consider the section $s(x) = x_0 - \beta x_1$ of $O(1)$. We then set $\overline{L}_{\beta}$ be to be the adelic line bundle where for a place $v \in M_K$, we have the metric $\log ||s(x)||_{v,\beta}^{-1} = \lambda_v(x,\beta)$. For any non-archimedean place $v$, we have 
$$||s(x_0:x_1)||_{v,\beta} = \frac{|x_0 - \beta x_1|_v}{\max\{|x_0|_v,|x_1|_v\} \max\{1, |\beta|_v\}}$$
and so if $|\beta|_v = 1$, we obtain the standard metric. Thus we get an adelic line bundle. The corresponding height function is then given by
$$h_{\beta}(x) = \frac{1}{|F_x|} \sum_{y \in F_x} \sum_{v \in M_K} N_v \log ||s(y)||_{v,\beta}^{-1} = \frac{1}{|F_x|} \sum_{y \in F_x} \sum_{v \in M_K} N_v \lambda_v(y,\beta).$$
This extends to a height function for any $\operatorname{Gal}(\overline{K}/K)$-invariant set $F$.
\par 
By Theorem \ref{Pairing1}, as $h_{\varphi}(F_n) \to 0$, we have $h_{\overline{L}_{\beta}}(F_n) \to \langle \overline{L}_{\beta}, \overline{L}_{\varphi} \rangle$. But by design, we have $h_{\overline{L}_{\beta}}(F_n) = h_{\beta}(F_n)$. Hence as $n \to \infty$, by Theorem \ref{Pairing1} we have $h_{\beta}(F_n) \to \langle \overline{L}_{\beta}, \overline{L}_{\varphi} \rangle$. Now expanding the definition of the pairing by using the section $s(x) = x_0 - \beta x_1$, we have
$$\langle \overline{L}_{\beta}, \overline{L}_{\varphi}  \rangle = h_{\varphi}(\beta) + \sum_{v \in M_K} N_v \int_{\mathbf{P}^1(\mathbb{C}_v)} \log ||s(x)||^{-1}_{v,\beta} d \mu_{\varphi,v}(x).$$
We thus obtain the following proposition.

\begin{proposition} \label{Pairing3}
Let $F_n$ be a sequence of $\operatorname{Gal}(\overline{K}/K)$-invariant sets such that $h_{\varphi}(F_n) \to 0$. Then for any $\beta \in \mathbb{P}^1(K)$, we have
$$h_{\beta}(F_n) \to h_{\varphi}(\beta) + \sum_{v \in M_K} N_v \int \lambda_v(x,\beta) d \mu_{\varphi,v}(x).$$
\end{proposition}

For each place $v$, by standard equidistribution theorems we have weak convergence of measures $\frac{1}{|F_n|}\sum_{x \in F_n} \delta_{x} \to \mu_{\varphi}$ in $\mathbb{P}^1_{\operatorname{Berk},v}$. We now attempt to enlarge the space of test functions. 

\begin{definition}\label{LogDefn1}
Let $(F_n)$ be a sequence of $\operatorname{Gal}(\overline{K}/K)$-invariant sets such that $h_{\varphi}(F_n) \to 0$. We say that $(F_n)$ satisfies logarithmic equidistribution at $\beta$ if 
$$\lim_{n \to \infty} \frac{1}{|F_n|} \sum_{x \in F_n} \lambda_v(x,\beta) \to \int \lambda_v(x,\beta) d \mu_{\varphi,v}.$$
\end{definition}

We then have the following theorem.

\begin{theorem} \label{LogEquibSIntegral1}
Let $(F_n)$ be a sequence of $\operatorname{Gal}(\overline{K}/K)$-invariant sets such that $h_{\varphi}(F_n) \to 0$. Furthermore assume that $(F_n)$ satisfies logarithmic equidistribution at $\beta$. Let $S$ be a finite set of places including the archimedean places. Then 
$$\lim_{n \to \infty} \left[ h_{\beta}(F_n) - \frac{1}{|F_n|}\sum_{v \in S}  \sum_{x \in F_n} N_v \lambda_v(x,\beta)  \right] \geq h_{\varphi}(\beta).$$
In particular, if $\beta$ is not preperiodic then there are only finitely many $F_n$'s that are $S$-integral with respect to $\beta$.
\end{theorem}

\begin{proof}
Since $h_{\varphi}(F_n) \to 0$, by Proposition \ref{Pairing3}, we have

$$h_{\beta}(F_n) \to h_{\varphi}(\beta) + \sum_{v \in M_K} N_v \int \lambda_v(x,\beta) d \mu_{\varphi,v}(x).$$
Now as $(F_n)$ satisfies logarithmic equidistribution at $\beta$, we also have
$$\lim_{n \to \infty} \frac{1}{|F_n|}\sum_{x \in F_n} N_v \lambda_v(x,\beta)  =N_v \int \lambda_v(x,\beta) d \mu_{\varphi,v}.$$
Thus
$$\lim_{n \to \infty} \left[ h_{\beta}(F_n) - \frac{1}{|F_n|}\sum_{v \in S} \sum_{x \in F_n} N_v \lambda_v(x,\beta) \right] $$
$$= h_{\varphi}(\beta) + \sum_{v \in M_K \setminus S} N_v \int_{\mathbb{P}^1_{\operatorname{Berk},v}} \lambda_v(x,\beta) \mu_{\varphi}(x) \geq h_{\varphi}(\beta)$$
as $\lambda_v(x,\beta)\geq 0$. In particular, if $\beta$ is non-preperiodic, then $h_{\varphi}(\beta) > 0$ and so for all large $n$ we must have
$$h_{\beta}(F_n) - \frac{1}{|F_n|} \sum_{v \in S} \sum_{x \in F_n} N_v \lambda_v(x,\beta) > 0$$
In particular, there must exist $v \in S$ such that $\sum_{x \in F_n} \lambda_v(x,\beta) > 0$ and thus $F_n$ is not $S$-integral with respect to $\beta$ as desired. 
\end{proof}

\subsection{Convergence to the Arakelov--Zhang Pairing} In this subsection, We apply our quantitative equidistribution theorem to the Arakelov--Zhang pairing. Let $\overline{L}_{\beta}$ be the adelic line bundle as in 4.2, where the metric on the section $s(x) = x_0 - \beta x_1$ gives rise to $\lambda_v(x,\beta)$ for the place $v$. For a rational function $\varphi$, let $\overline{L}_{\varphi}$ be the adelic line bundle equipped with the canonical metric $|| \cdot ||_{\varphi,v}$ for each place $v$. Let $f_v = \lambda_v(x,\beta) - \log ||s(x)||_{\varphi,v}$. We wish to estimate the rate of convergence for $f_v$. 
\par 
We observe that $\triangle f_v = \zeta(0,1) - \mu_{\varphi,v}$, and so differs from the dynamical Green's function based at $\zeta(0,1)$ by a constant. In particular, it is independent of $\beta$ and only depends on the place $v$. Thus it is Hölder continuous for some exponent $\kappa'$ independent of $\beta$. Applying Proposition \ref{QuantEquib2} with $\kappa = 2 \kappa'$ and combining constants, we obtain 

\begin{proposition} \label{AraZhEquib1}
There exists $C > 0$ such that for any $\operatorname{Gal}(\overline{K}/K)$-invariant set $F$, we have
$$\left| \frac{1}{|F|} \sum_{x \in F} f_v(x) - \int f_v(x) d\mu_{\varphi,v} \right| \leq C\left(\frac{1}{|F|^{1/2}} + (h_{\varphi}(F) +  \frac{\log |F|}{|F|})^{1/2} \right).$$
\end{proposition}

We now obtain a bound on the rate of convergence of the Arakelov--Zhang pairing, where fix the rational map $\varphi$ but allow $\beta$ is allowed to be in $\overline{K}$. 

\begin{proposition} \label{AraZhEquib2}
Let $\varphi$ be a rational map defined over $K$ of degree $d \geq 2$. There exists $C_{AZ,\varphi} > 0$ depending only on $\varphi$ such that for all $\beta \in \overline{K}$, 
$$|h_{\overline{L}_{\beta}}(F) - \langle \overline{L}_{\beta}, \overline{L}_{\varphi} \rangle| \leq h_{\varphi}(F) + C_{\operatorname{AZ},\varphi}   (h_{\varphi}(F)^{1/2} + \frac{1 + \log |F|^{1/2}}{|F|^{1/2}} ).$$
\end{proposition}

\begin{proof}
Let $K'$ be an extension of $K$ that contains $\beta$ and let $S$ be the finite set of places of $K$ where $\mu_{\varphi,v} \not = \delta_{\zeta(0,1)}$. Recall from Corollary \ref{Pairing2} that for any place $v \in M_K$, we have the function 
$$f_v = \frac{ \log || s(x)||_{v,\epsilon,\varphi}}{\log || s(x)||_{v}}$$ 
where $\epsilon$ is some polarization for $\varphi$, whose equidistribution rate controls our bound for the Arakelov--Zhang pairing. Then for any place $v \not \in S$ and $w \in M_{K'}$ that extends $v$, we have that $f_w$ is constant. Thus by Corollary \ref{Pairing2}, we have
$$| h_{\overline{L}_{\beta}}(F) - \langle \overline{L}_{\beta}, \overline{L}_{\varphi} \rangle| \leq h_{\varphi}(F) + \sum_{v \in S} \sum_{w \mid v} N_w \left|\frac{1}{|F|} \sum_{x \in F} f_w(x) - \int f_w(x) d \mu_{\varphi,w} \right|.$$
Since $f_w(x) = f_v(x)$, we can use Proposition \ref{AraZhEquib1} to obtain a constant $C > 0$ such that
$$|h_{\overline{L}_{\beta}}(F) - \langle \overline{L}_{\beta} , \overline{L}_{\varphi} \rangle| \leq h_{\varphi}(F)  + C \left(\frac{1}{|F|^{1/2}} + (h_{\varphi}(F) + \frac{\log |F|}{|F|})^{1/2} \right) \sum_{v \in S} \sum_{w \mid v} N_w.$$
Now we note that $\sum_{v \in S} \sum_{w \mid v} N_w = \sum_{v \in S} N_v$ and thus is a constant independent of our extension $K'$. Using the inequality $(x+y)^{1/2} \leq x^{1/2} + y^{1/2}$, we get what we desired.
\end{proof}




\section{Linear Forms in Logarithms}
In this section, we collect the relevant theorems from diophantine approximation that will be needed to prove our main theorems. The main tool that we will be needing is linear forms in logarithms. This was first developed by Baker (see \cite{Bak75}) that allowed one to give lower bounds to quantities of the form $|a_1^{b_1} \cdots a_n^{b_n} - 1|$ in terms of the heights of $a_i$ and $b_i$. 
\par 
The first result that we need is a theorem of Laurent, Mignotte and Nesterenko.


\begin{theorem} \label{Baker} (Theorem 3 \cite{LMN95})
Let $\alpha$ be an algebraic number with $|\alpha| = 1$ which is not a root of unity and let $b_1,b_2$ be positive integers. Define $\Lambda = b_1 i \pi - b_2 \log \alpha$. Let
$$D = [\mathbb{Q}(\alpha):\mathbb{Q}]/2, \quad a = \max \{20, 10.98|\log \alpha| + D h(\alpha) \},$$
$$H = \max\left \{17, \frac{\sqrt{D}}{10}, D \log(\frac{b_1}{2a} + \frac{b_2}{68.9}) + 2.35D + 5.03 \right\}.$$
Then
$$ \log |\Lambda| \geq -8.87aH^2.$$
\end{theorem}

\begin{corollary} \label{BakerCorollary1}
Let $\beta$ be an algebraic number with $|\beta| = 1$. Then for any $\epsilon > 0$, there exists a constant $C_{\epsilon} > 0$ such that for any root of unity $\zeta$ of order $n$, we have
$$\log |\beta - \zeta| \geq -C_{\epsilon} [\mathbb{Q}(\beta):\mathbb{Q}]^3 (h(\beta)+1) n^{\epsilon}.$$
\end{corollary}

\begin{proof}
Since $\beta$ and $\zeta$ are both on the unit circle, it suffices to bound how close their angles are. Thus we wish to bound the quantity $\log |\frac{k}{n} \pi i  - \log \beta|$. Since Theorem \ref{Baker} requires $b_1,b_2$ to be integers, we will instead bound $\log |k \pi i - n \log \beta|$ which incurs at most an extra $\log n$. As we may take $|\log \beta|$ to be less than $2 \pi$, and so we can take $a$ to be $ O([\mathbb{Q}(\beta):\mathbb{Q})] (h(\beta)+1))$. For $H$, as $b_1, b_2 = O(n)$, we can take $H$ to be $O(D \log n)$. Thus we can find a constant $C > 0$ such that
$$\log | k \pi i - n \log \beta| \geq -C [\mathbb{Q}(\beta):\mathbb{Q}]^3 (h(\beta)+1) (\log n)^2.$$
Since $\log n << n^{\epsilon}$ for all large enough $n$, we obtain that 
$$\log \left| \frac{k}{n} \pi i - \log \beta \right| \geq - C_{\epsilon} [\mathbb{Q}(\beta):\mathbb{Q}]^3 (h(\beta)+1) n^{\epsilon}$$
as desired.
\end{proof}

We now move onto the case of elliptic curves. We will require a version of linear forms on logarithms that handle points on elliptic curves. Let $E$ be an elliptic curve and fix an isomorphism $\mathbb{C}/\Lambda \simeq E(\mathbb{C})$, where $\Lambda$ is the period lattice. This map is given by $z \mapsto [\wp(z) : \wp'(z) : 1]$ where $\wp(z)$ is the Weierstrass p-function associated to the lattice $\Lambda$. If we let $\omega_1,\omega_2$ be a basis for the lattice $\Lambda$ and $\Pi$ a fundamental domain for $\mathbb{C}/\Lambda$, then we may view our isomorphism $\Pi \simeq E(\mathbb{C})$ as an exponential map and we denote the inverse map by $\log_E$. We now state a theorem due to David--Hirata-Kohno with some simplifications done. For an elliptic curve $E$, we let $h_E$ denote the Neron--Tate height on it.

\begin{theorem}[Theorem 1, \cite{DHK02}] \label{EllipticBaker1}
Let $k$ be a positive integer and let $E$ be an elliptic curve defined over some number field $K$. There exists a constant $C > 0$ such that the following hold: Let $K'$ be a number field of degree $D$ over $K$ and $L(z) = \beta_0 z_0 + \cdots + \beta_k z_k$ be a non-zero linear form on $\mathbb{C}^{k+1}$ with coefficients in $K'$. Let $u_1,\ldots,u_k$ be complex numbers such that $\gamma_i = (1,\wp(u_i), \wp'(u_i)) \in E(K')$. Let $B,V_1,\ldots,V_k$ be real numbers satisfying
$$\log B \geq \max\{1, h(\beta_i); 0 \leq i \leq k \}$$
$$V_1 \geq \cdots \geq V_k$$
$$\log V_i \geq \max \left \{ e, h_{E}(\gamma_i), \frac{|u_i|^2}{D} \right \}.$$
Then if $L(v) \not = 0$ for $v = (1,u_1,\ldots,u_k)$, we have
$$\log |L(v)| \geq -C D^{2k+2} (\log B + \log(eD) + \log \log V_1)(\log (eD) + \log \log V_1)^{k+1} \prod_{i=1}^{k} (1 + \log V_i).$$
\end{theorem}

\begin{corollary} \label{EllipticBaker2}
Let $E$ be an elliptic curve defined over a given number field $K$. Then there exists a constant $C > 0$ for which the following holds: Let $L/K$ be an extension of $K$ with degree $D$. Let $b_1,b_2$ a rational numbers and $\alpha$ an element of $E(L)$. Let 
$$\Lambda =  b_1  + b_2 \omega + \log_E \alpha$$
where our elliptic curve $E$ is given by the period lattice spanned by $\{1, \omega\}$. Let $B > 0$ be a constant such that $\log B > \max\{1,h(b_1),h(b_2)\}$. Then if $\Lambda \not = 0$, we have
$$\log |\Lambda| \geq -C D^6 (\log D + 1)^2 \log B (h_E(\alpha)+1) (\log^+ h_E(\alpha)+1)^4.$$
\end{corollary}

\begin{proof}
We apply Theorem \ref{EllipticBaker1} with $k = 2$. Then $\log V_1$ may be taken as $O(1)$, and $\log V_2$ taken as $O(h_E(\alpha))$ as $|u_i|$'s are bounded. The result then follows immediately. 
\end{proof}

Observe that there is an extra $\log^+ h_E(\alpha)$ factor in Corollary \ref{EllipticBaker2}, as compared to Corollary \ref{BakerCorollary1}. This factor is the main obstruction in proving a uniform result like Theorem \ref{IntroUniformIh1} for general Lattès maps. For elliptic curves with complex multiplication, Ably and Graudon \cite{AG03} have managed to remove the $\log^+ h_E(\alpha)$ factor, which will allow us to prove Theorem \ref{IntroUniformIh2}. We now state a special case of their theorem that we need, rephrased in quantities that we use in Theorem \ref{EllipticBaker1}. 

\begin{theorem}[Theorem 0.1 \cite{AG03}] \label{EllipticBakerCM1}
Let $k$ be a positive integer and let $E$ be an elliptic curve with complex multiplication that is defined over some number field $K$. There exists a constant $C > 0$ such that the following hold: Let $K'$ be a number field of degree $D$ over $K$ and $L(z) = \beta_0 z_0 + \cdots + \beta_k z_k$ be a non-zero linear form on $\mathbb{C}^{k+1}$ with coefficients in $K'$. Let $u_1,\ldots,u_k$ be complex numbers such that $\gamma_i = (1,\wp(u_i), \wp'(u_i)) \in E(K')$. Let $B,V_1,\ldots,V_k$ be real numbers satisfying
$$\log B \geq \max\{1, h(\beta_i); 0 \leq i \leq k \}$$
$$V_1 \geq \cdots \geq V_k$$
$$\log V_i \geq \max \{ e, h_{E}(\gamma_i), \frac{|u_i|^2}{D} \}.$$
Then if $L(v) \not = 0$ for $v = (1,u_1,\ldots,u_k)$, we have
$$\log |L(v)| \geq -C (D^n (1 + D \log (1+D) \prod_{i=1}^{n} \log V_i))$$
$$\times (D \log B + \log (e + \max_{1 \leq i \leq n} |u_i| + \log \max_{1 \leq i \leq n} \{1 , \frac{1}{|u_i|} \})).$$
\end{theorem}


\begin{corollary} \label{EllipticBakerCM2}
Let $E$ be an elliptic curve with complex multiplication that is defined over a number field $K$ and let $D$ be a positive integer. Then there exists a constant $C$ such that the following holds: Let $b_1,b_2$ be two rational numbers and $\alpha$ an element of $E(L)$ where $[L:K] \leq D$. Let 
$$\Lambda =  b_1  + b_2 \omega + \log_E \alpha$$
where our elliptic curve $E$ is given by the period lattice spanned by $\{1, \omega\}$. Let $B > 0$ be a constant such that $\log B > \max\{1,h(b_1),h(b_2)\}$. Then if $\Lambda \not = 0$, we have
$$\log |\Lambda| \geq -C (h_E(\alpha)+1) \log B.$$
\end{corollary}

\begin{proof}
This follows immediately from Theorem \ref{EllipticBakerCM1}, where now since we fix an upper bound of the degree $D$ and do not care about the dependency on $D$, we can treat $D$ as a constant. Then we may choose $|u_i|$ to be some bounded domain away from the origin, so that both $|u_i|$ and $\frac{1}{|u_i|}$ are bounded. We are then left with the $\prod_{i=1}^{n} \log V_i$ term, which can be chosen to be $O(\log_E h(\alpha))$ as desired.
\end{proof}



\section{Uniform Ih's Conjecture for $\varphi(z) = z^d$ and Lattès maps}
Let $\varphi(z)$ be either $z^d$ or a Lattès map. In this section, we will establish Theorems \ref{IntroUniformIh1}, \ref{IntroUniformIh3} and \ref{IntroUniformIh2}. In the case of Lattès maps, let's first make some reductions. Recall that a Lattès map is a rational map $\varphi: \mathbb{P}^1 \to \mathbb{P}^1$ that fits into a commutative diagram

$$
\begin{tikzcd}
E \arrow{r}{\psi} \arrow{d}[swap]{\pi}  & E \arrow{d}[swap]{\pi}  \\
\mathbb{P}^1 \arrow{r}{\varphi} & \mathbb{P}^1
\end{tikzcd}
$$

\noindent where $\pi: E \to \mathbb{P}^1$ is a non-constant map and $\psi: E \to E$ is a self map (not necessarily a group homomorphism) of degree $d \geq 2$. For a given Lattès map $\varphi$, there are multiple possible pairs $(E,\pi)$ that make the above diagram commute, but it is possible for us to make a convenient choice using the following proposition.


\begin{proposition} \label{LattèsRepresentative} [Theorem 6.57 \cite{Sil07}] 
Every Lattès map fits into a diagram of the following form
$$
\begin{tikzcd}
E \arrow{r}{\psi} \arrow{d}[swap]{\pi}  & E \arrow{d}[swap]{\pi}  \\
\mathbb{P}^1 \arrow{r}{\varphi} & \mathbb{P}^1
\end{tikzcd}
$$
where $\pi \in \{x,y,x^2,x^3\}$ where $x$ and $y$ are the projection to the $x$ or $y$-coordinates for some choice of Weierstrass equation for $E$.
\end{proposition}

Furthermore, we may choose the elliptic curve $E$ to be the same elliptic curve as part of the data of our Lattès map. In particular if we knew it came from an elliptic curve $E$ with complex multiplication, then we may also choose the elliptic curve $E$ in Proposition \ref{LattèsRepresentative} to also have complex multiplication. 
\par 
As $x^k$ being $S$-integral relative to $\beta$ implies that $x$ is $S$-integral relative to $\beta^{1/k}$, we may henceforth assume that $\pi = x$ or $y$ at the cost of assuming that the degree of $\beta$ is larger by some constant factor. Thus all Lattès maps that we consider from now on are assumed to be able to fit into a diagram with $\pi: E \to \mathbb{P}^1$ being either $\pi = x$ or $y$.

\subsection{Quantitative Logarithmic Equidistribution}

As shown in \cite{BIR08}, for $\varphi(z) = z^d$ or a Lattès map along with any sequence of distinct preperiodic points and $\beta \in \mathbb{P}^1(K)$, we have logarithmic equidistribution at $\beta$. Thus by Theorem \ref{LogEquibSIntegral1}, if $h_{\varphi}(\beta) > 0$ then there are only finitely many preperiodic points that are $S$-integral relative to $\beta$. 
\par 
To obtain a uniform statement, for a fixed place $v \in M_K$, we will have to give an upper bound for the expression
$$\left|\frac{1}{|F|} \sum_{x \in F} \lambda_v(x,\beta) - \int \lambda_v(x,\beta) d\mu_{\varphi,v} \right|$$
in terms of $h(\beta)$ and $|F|$, where $F$ is the Galois orbit of some preperiodic point. If we can show the existence of a constant $c > 0 $ such that the error is bounded by $ch(\beta)/|F|$, then for $|F|$ large enough we would have
$$\left| \frac{1}{|F|}\sum_{v \in S} \sum_{x \in F} N_v \lambda_v(x,\beta) - \sum_{v \in S} \int N_v \lambda_v(x,\beta) d \mu_{\varphi,v} \right| < \frac{1}{2} h(\beta).$$
But by the Arakelov--Zhang pairing, we must have
$$\lim_{|F| \to \infty} \frac{1}{|F|} \sum_{v \in S} \sum_{x \in F} N_v \lambda_v(x,\beta) \to h_{\varphi}(\beta) + \sum_{v \in M_K} \int N_v \lambda_v(x,\beta) d \mu_{\varphi,v}$$
and so if $h_{\varphi}(\beta) > \frac{1}{2} h(\beta)$, which is true for $\beta$ of large height, we obtain that $F$ cannot be $S$-integral relative to $\beta$. We can then take $n$ large enough to handle all points of small height, of which there are only finitely many of them.
\par 
We first study the situation in general, before specializing to the case of $z^d$ and Lattès maps. Let's say we have an upper bound on how close $x \in F$ can be to $\beta$, say $\log |x-\beta|_v^{-1} < C_1$. We will first estimate the rate of equidistribution for the function $\log^+|x-\beta|_v^{-1}$, as it turns out to be easier, before turning to $\lambda_v(x,\beta)$. 
\par 
Define $f_M = \min\{\log M, \log^+|x-\beta|_v^{-1})$. Then $f_M$ is a continuous function, with $\triangle f_M =  \zeta_{\beta,1} - \zeta_{\beta,M}$. It is Lipschitz continuous, with a constant given by $M$ as the derivative of $\log |x|$ is upper bounded by $\frac{1}{|x|}$. Applying Proposition \ref{QuantEquib2}, we get that
$$\left| \frac{1}{|F|} \sum_{x \in F} f_M(x) - \int f_M d\mu_{\varphi,v} \right| \leq \frac{M}{|F|^{1/\kappa}} + C \left(\frac{\log |F|}{|F|} \log M \right)^{1/2}$$
for $\kappa$ a Hölder exponent of the equilibrium measure $\mu_{\varphi,v}$. Now since $\log |x-\beta|^{-1}_v < C_1$ for $x \in F$, it follows that
$$\left| \frac{1}{|F|} \sum_{x \in F} f_M(x) - \frac{1}{|F|}\sum_{x \in F} f_M(x) \right| \leq \frac{C_1}{|F|} \left|F \cap D(\beta,\frac{1}{M}) \right|.$$
Using Proposition \ref{QuantDisc1} and the fact that $\mu_{\varphi,v}(D(x, \epsilon)) = O(\epsilon^{\kappa})$ due to Hölder-continuous potentials, we get
$$\left| \frac{1}{|F|} \sum_{x \in F} f_M(x) - \frac{1}{|F|}\sum_{x \in F} f(x) \right| \leq C_1 \left(\frac{M}{|F|^{1/\kappa}} + O\left(\frac{1}{M^{\kappa}}\right) + O\left(\frac{\sqrt{\log |F|}}{\sqrt{|F|}} \right) \right).$$
Using the fact that it has Hölder-continuous potentials again, we may estimate that
$$\left| \int f_M d\mu_{\varphi,v} - \int \log^+|x-\beta|^{-1}_v d \mu_{\varphi,v} \right| \leq O\left(\int_0^{M} \log x dx^{\kappa} \right) = O \left(\frac{\log M}{M} \right).$$
Putting it together and choosing $\kappa < 1/4$ with $M = |F|^{1/2 \kappa}$, we obtain
$$\left|\frac{1}{|F|} \sum_{x \in F} \log^+|x-\beta|^{-1}_v - \int \log^+|x-\beta|^{-1}_v \right| \leq C_1 O\left(\frac{\sqrt{\log |F|}}{\sqrt{|F|}} \right) + O \left(\frac{\sqrt{\log |F|}}{{\sqrt{|F|}}} \right)$$
where both big $O$ constants depends only on our rational map $\varphi$ and not on $\beta$. If we assume that $C_1 > 1$, then we can combine the two expressions. The above discussions then implies that

\begin{proposition} \label{GeneralQuantLog1}
Let $v \in M_K$ be a place of $K$ and $\beta \in \mathbb{P}^1(K)$ be a point. Fix some $\delta$ with $\frac{1}{2} > \delta > 0$ and a rational map $\varphi$ over $K$. Then there exists a constant $C = C(\varphi,\delta) > 0$ such that for any $A> 1$, if 
$$
\max_{x \in F} \log|x-\beta|^{-1}_v < A(h(\beta) +1)|F|^{1/2 - \delta}
$$
for some $\operatorname{Gal}(\overline{K}/K)$-invariant set $F$ of preperiodic points, then 
$$\left| \frac{1}{|F|} \sum_{x \in F} \log^+|x-\beta|^{-1}_v - \int \log^+|x-\beta|^{-1}_v d \mu_{\varphi,v} \right| \leq \frac{C}{|F|^{\delta}} \sqrt{\log |F|} A (h(\beta)+1).$$
\end{proposition}

We now derive a similar bound for $\lambda_v(x,\beta)$.

\begin{lemma} \label{GeneralQuantLog2}
Let $v \in M_K$ be a place of $K$ and $\beta \in \mathbb{P}^1(K)$ be a point. Fix some $\delta$ with $\frac{1}{2} > \delta > 0$ and a rational map $\varphi$ over $K$. Then there exists a constant $C = C(\varphi,\delta) > 0$ such that for any $A> 0 $, if 
$$\max_{x \in F} \log|x-\beta|^{-1}_v < A (h(\beta) +1)|F|^{1/2 - \delta}$$
for some $\operatorname{Gal}(\overline{K}/K)$-invariant set $F$ of preperiodic points, then 
$$\left| \frac{1}{|F|} \sum_{x \in F} \lambda_v(x,\beta) - \int \lambda_v(x,\beta) d \mu_{\varphi,v} \right| \leq \frac{C}{|F|^{\delta}} \sqrt{\log |F|} A (h(\beta)+ \log^+|\beta|_v + 1).$$
\end{lemma}

\begin{proof}
Let $f_{\beta} = \lambda_v(x,\beta) - \log^+|x-\beta|_v^{-1}$. It suffices to find a constant $C' > 0$ such that
$$\left| \frac{1}{|F|} \sum_{x \in F} f_{\beta}(x)  - \int f_{\beta}(x) d \mu_{\varphi,v} \right| \leq \frac{C'}{|F|^{\delta/2}}(\log^+|\beta|_v + 1).$$
Our function $f_{\beta}(x)$ is a continuous function whose Laplacian equals $\zeta(\beta,1) - \zeta(0,1)$. One can then check that $(\triangle f_{\beta}, \triangle f_{\beta}) = O(\log^+|\beta|_v + 1)$. Next, we have to determine a Lipschitz constant. Writing $\lambda_v(x,\beta) = \log^+|x|_v + \log^+|\beta|_v -\log |x-\beta|_v$, we have 
$$f_{\beta}(x) = \log^+|x|_v + \log^+|\beta|_v - \log |x-\beta|_v - \log^+|x-\beta|^{-1}_v.$$
The term $\log^+|x|_v$ has a Lipschitz constant of at most one and $\log^+|\beta|_v$ is a constant. When $x \in D(\beta,1)$, the last two terms cancel out and we have a Lipschitz constant of one. When $x$ is not in it, the term $\log^+|x-\beta|^{-1}_v = 0$ and $\log |x-\beta|_v$ has a Lipschitz constant of one too, giving us $2$ as an upper bound. Applying Proposition \ref{QuantEquib2}, we get
$$\left| \frac{1}{|F|} \sum_{x \in F} f_{\beta}(x)  - \int f_{\beta}(x) d \mu_{\varphi,v} \right| < \frac{2}{|F|^{1/\kappa}} + C \frac{(\log |F|)^{1/2}}{|F|^{1/2}} (\log^+|\beta|_v + 1)^{1/2}$$
for some $C > 0$ depending only on $\varphi$. Since $\delta < 1/2$, this implies what we need. 
\end{proof}


\subsection{Bounding the distance to preperiodic points for power maps}

Our aim now is to prove the existence of a constant $A$ for the inequality
\begin{equation} \label{Bound}
\max_{x \in F} \log|x-\beta|^{-1}_v < A (h(\beta) +1)|F|^{1/2 - \delta} \tag{*}
\end{equation}

\noindent 
that is a power of the degree $[K(\beta):K]$ for both $\varphi(z) = z^d$ and Lattès map. As noted in the introduction, it will be necessary for us to allow a finite number of exceptions. We first handle the case of $\varphi(z) = z^d$ first. 

\begin{proposition} \label{PowerBound1}
Let $\varphi(z) = z^d$. For any $\epsilon > 0$, there exists a constant $C_{\epsilon}$ such that the following holds: Let $K$ be a number field and $\beta \in \mathbb{P}^1(K)$  be a non-preperiodic point. Let $v$ be an archimedean place of $K$. Then for any $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-orbit $|F|$ of preperiodic points, we have
$$\max_{x \in F} \log|x-\beta|^{-1}_v < C_{\epsilon}[K:\mathbb{Q}]^3(h(\beta)+1) |F|^{\epsilon}.$$
\end{proposition}

\begin{proof}
We first handle the case where $\beta$ does not lie on the unit circle. Then the distance to any root of unity can be bounded from below by $|\beta|_v - 1$. Now $|\beta|_v$ is a real algebraic number living in some field $K'$ of degree at most $2$ larger than $K$. Since $\beta \overline{\beta} = |\beta|_v$ where $\overline{\beta}$ is the complex conjugate, the height of $|\beta|_v$ is at most twice of that of $\beta$. By Proposition 5(b) of \cite{HS11}, we have $h(|\beta|_v - 1) \leq h(|\beta|_v) + \log 2$ and so $h(|\beta|_v - 1) \leq 2h(\beta) + \log 2$. Hence we conclude that
$$ \log | |\beta|_v - 1|^{-1} \leq [K':\mathbb{Q}](2h(\beta) + \log 2)$$ 
which gives us the constant $C$ we need.
\par 
Now let's us assume that $\beta$ lies on the unit circle and let $\zeta$ be a primitive $n^{th}$ root of unity. Applying Corollary \ref{BakerCorollary1}, as the Galois orbit $F$ of $\zeta$ has cardinality $\varphi(n) \geq \sqrt{n}$, we obtain a constant $C_{\epsilon} > 0$
$$\max_{x \in F} \log|x-\beta|^{-1}_v < C_{\epsilon}[K(\beta):K]^3 (h(\beta)+1)|F|^{\epsilon}$$
for any $\epsilon > 0$ as desired.
\end{proof}


\begin{proposition} \label{PowerBound2}
Fix a non-archimedean place $v$ of $\mathbb{Q}$ corresponding to the prime $p$, let $D$ be a positive integer and let $\delta > 0$. Then there exists a constant $C> 0$ such that  for any $\beta \in \mathbb{P}^1(K)$ with $[K:\mathbb{Q}] < D$ and root of unity $\zeta$, we have
$$\log|\zeta-\beta|_v^{-1} < \delta$$
if $\deg_{\mathbb{Q}}(\zeta) > C$.
\end{proposition}

\begin{proof}
As each root of unity lies in a different residue class unless their order differs by a power of $p$, for $\beta \in K$ we have $\lambda_v(\beta,\zeta) = 0$ unless $\operatorname{ord}(\zeta) = p^k a_i$ for some finite set $\{a_1,\ldots,a_n\}$ of integers coprime to $p$. We can choose the $a_i$'s such that it works for any field extension $K$ of degree $D$, as the inertia degree of such a field is at most $D$.
\par 
Let $\zeta_m$ denote a primitive $m^{th}$ root of unity. Let $|\zeta_{p^k} - 1|_v = c$ . If $\operatorname{ord}(\zeta) = p^k a_i$, we may write $\zeta = \zeta_{p^k}^a \zeta_{a_i}^b$ for some natural numbers $a,b$ and thus conclude that $1 > |\zeta - \zeta_{a_i}^b| = c$. Hence 
$$c > |\beta - \zeta|_v \implies |\beta - \zeta_{a_i}^b|_v = c.$$
Due to ramification, we have $c = 1/p^{k-1}(p-1)$ and since we have only finitely many $\zeta_{a_i}$'s, the element $\beta - \zeta_{a_i}^j$ all live in some number field of fixed degree and thus it is impossible for our distance to be $c$ if $k$ is large enough. Hence if $k$ is large enough, $|\beta - \zeta|_v \geq c$ and so
$$\log |\beta - \zeta|_v^{-1} < \frac{1}{p^{k-1}(p-1)} \log p < \delta.$$
Hence if the order of $\zeta$ is larger than some constant $C$, we have $\log |\zeta - \beta|^{-1}_v < \delta$ as desired.
\end{proof}

As a simple corollary of the proof, we deduce the following too.

\begin{corollary} \label{PowerBound3}
Let $\varphi(z) = z^d$ and fix a non-archimedean place $v$ of $\mathbb{Q}$ corresponding to the prime $p$. Then for any $\beta \in \mathbb{P}^1(\overline{\mathbb{Q}})$, there do not exist two distinct roots of unity $\zeta_1,\zeta_2$ such that
$$\log |\zeta_i - \beta|^{-1}_v > \log p.$$
\end{corollary}

\begin{proof}
The condition implies that $|\zeta_i - \beta|_v < \frac{1}{p}$ and so $|\zeta_1 - \zeta_2|_v < \frac{1}{p}$. Thus $|1 - \zeta_1^{-1} \zeta_2|_v  < \frac{1}{p}$, which is impossible as $|1 - \zeta|_v$ is at least $1/p^{p-1}$ for any root of unity $\zeta$.
\end{proof}

\subsection{Bounding the distance to preperiodic points for Lattès Maps}

We now move onto the case of Lattès maps. Recall from the discussion at the beginning of our section that we may assume our Lattès map fits into a commutative diagram

$$
\begin{tikzcd}
E \arrow{r}{\psi} \arrow{d}[swap]{\pi}  & E \arrow{d}[swap]{\pi}  \\
\mathbb{P}^1 \arrow{r}{\varphi} & \mathbb{P}^1
\end{tikzcd}
$$

where $\pi: E \to \mathbb{P}^1$ is either the $x$ or $y$-coordinate map when $E$ is an elliptic curve written in Weierstrass coordinates $y^2 = x^3 + ax + b$. 
\par 
Now let $\varphi$ be defined over a number field $K$ and let $v$ be an archimedean place. Then $\pi$ is either the $x$-coordinate function, which can be viewed as the Weierstrass $p$-function $\wp(z)$, or the $y$-coordinate function which can be viewed as the derivative $\wp'(z)$. 
\par 
To apply Corollary \ref{EllipticBaker2}, we have to control the distance between $\wp(z)$ and $\wp(w)$ in terms of $|z-w|$ where $z,w$ are points of $\mathbb{C}/\Lambda$. The Weierstrass $p$-function is a two-to-one map, with either two simple zeroes or a double zero. Locally around the zeros, the function $\wp(z) : \mathbb{C}/\Lambda \to \mathbb{P}^1$ is conjugate to $z^k$ for $k = 1$ or $2$ and so away from $\infty$ there exists a constant $C$ such that $|z-w| \leq C|\wp(z) - \wp(w)|^{1/2}$. At $\infty$, we have to invert the coordinate at $\mathbb{P}^1$ and thus we get $|z-w| \leq C|\frac{1}{\wp(z)} - \frac{1}{\wp(w)}|^{1/2}$. At other points, $\wp(z)$ is locally invertible and so locally we have a constant $C$ such that $|z-w| \leq C|\wp(z)-\wp(w)|$. By compactness, we conclude the following proposition.

\begin{proposition} \label{DistanceBound}
There exists constants $\delta > 0$ and $C > 0$ such that if $x,y \in \mathbb{C}$ with $|x-y| < \delta$, then there exists $z,w \in \mathbb{C}/\Lambda$ such that $\wp(z) = x, \wp(w) = y$ and $|z-w| \leq C |x-y|^{1/2} \max\{|x|^{1/2},1\}$.
\end{proposition}

Similarly, we obtain the following proposition for $\wp'(z)$ where we have an exponent of $1/3$ as $\wp'(z) : \mathbb{C}/\Lambda \to \mathbb{P}^1$ is a $3:1$ map. 

\begin{proposition} \label{DistanceBound2}
There exists constants $\delta > 0$ and $C > 0$ such that if $x,y \in \mathbb{C}$ with $|x-y| < \delta$, then there exists $z,w \in \mathbb{C}/\Lambda$ such that $\wp'(z) = x, \wp'(w) = y$ and $|z-w| \leq C |x-y|^{1/3} \max\{|x|^{1/3},1\}$.
\end{proposition}



Hence to bound the distance between $x \in \mathbb{P}^1(K)$ and a preperiodic point for $\varphi$, we can bound the distance between a point $\alpha \in E(K')$ with $\wp(\alpha) = x$ and a torsion point. We first handle the case where $E$ has complex multiplication. 

\begin{proposition} \label{EllipticCMBound1}
Let $\varphi$ be a Lattès map associated to an elliptic curve $E$ defined over $K$, with complex multiplication and let $v$ be an archimedean place of $K$ and let $D$ be a positive integer. Then for any $\epsilon > 0$, there exists a constant $C_{\epsilon}$ such that for any $\beta \in \mathbb{P}^1(\overline{K})$ with $\deg_K(\beta) \leq D$ and any $\operatorname{Gal}(\overline{K}/K)$-orbit $F$ of preperiodic points, we have
$$\max_{x \in F} \log |x-\beta|_v^{-1} \leq C_{\epsilon} (h(\beta)+1)|F|^{\epsilon}$$
\end{proposition}

\begin{proof}
By Proposition \ref{DistanceBound} and \ref{DistanceBound2}, and noting that $h(x) > \frac{1}{D} \log^+ |x|^{1/2}$ for $|x| \geq 1$, it suffices to prove the existence of $C > 0$ such that
$$\max_{x \in F} \log | \log_E \alpha - \log_E x|^{-1}_v < C (h(\alpha) + 1) |F|^{\epsilon}$$
where $\alpha$ is a point in $E(\overline{K})$ such that $\deg_K(x(\alpha)) \leq D$. In particular, $\deg_K(\alpha) \leq 2D$. Let $n$ be the order of $x$ as a torsion point. Then $\log_E x$ can be written as $\frac{a_1}{n} + \frac{a_2}{n} \omega$ for some positive integers $a_1,a_2$. Applying Corollary \ref{EllipticBakerCM2}, we obtain that
$$\log |\log_E \alpha - \log_E x|^{-1}_v < C (h(\alpha)+1) \log n.$$
Now it suffices to note that $|F| > \frac{cn}{(\log \log n)^2}$ by say (45) and (46) of \cite{BIR08}, and we can replace $\log n$ with $|F|^{\epsilon}$ as desired.
\end{proof}



For a general Lattès map, applying Theorem \ref{EllipticBaker1} directly is not enough to prove the bound we need, as there is an extra factor of $\log^+ h(\alpha)$. Instead, we will exploit the fact that torsion points cannot be too close to each other.

\begin{proposition} \label{EllipticBound1}
Let $\varphi$ be a Lattès map defined over a number field $K$ and let $v$ be an archimedean place of $K$. Then for any $\epsilon > 0$, there exists a constant $C_{\epsilon}$ such that the following holds: for all $\beta \in \mathbb{P}^1(\overline{K})$ with degree $\deg_K(\beta) \leq D$, we have
$$\max_{x \in F} \log |x-\beta|^{-1}_v \leq C_{\epsilon}D^6 (\log D)^2 (h(\beta)+1)|F|^{\epsilon}$$
for all $\operatorname{Gal}(\overline{K}/K)$-orbits $F$ of preperiodic points with the possible exception of one.
\end{proposition}


\begin{proof}
Again by Proposition \ref{DistanceBound}, and noting that $h(x) > \frac{1}{D} \log^+ |x|^{1/2}$, it suffices to prove the existence of $C > 0$ such that
\begin{equation}\label{eq:Elliptic1} 
\max_{x \in F} \log | \log_E \alpha - x|^{-1}_v < C D^6 (\log D)^2 (h(\alpha) + 1) |F|^{\epsilon} 
\end{equation}
where $\alpha$ is a point in $E(\overline{K})$ with $\wp(\alpha) \in \mathbb{P}^1(K)$. By Corollary \ref{EllipticBaker2}, if we write $x$ as $b_1 \omega_1 + b_2 \omega_2$, then we have the bound
$$\max_{x \in F} \log |\log_E \alpha - x|^{-1}_v < C (\log B) D^4 (\log D)^2 (h_E(\alpha)+1) (\log^+ h_E(\alpha)+1)^4.$$
where $\log B = \max\{h(b_1),h(b_2)\}$. If $n$ is the order of $x$ as a torsion point, then $\log B = \log n$, and again by (45) and (46) of \cite{BIR08}, we know that $|F|  > \frac{cn}{(\log \log n)^2}$ for some constant $c > 0$. Thus if $n > h_E(\alpha)^{2/\epsilon}$, it would follow that
\begin{equation} \label{eq:Elliptic2}
\max_{x \in F} \log |\log_E\alpha - x|^{-1}_v < C D^6 (\log D)^2 (h_E(\alpha)+1) |F|^{\epsilon}
\end{equation}
for an appropriate constant $C$. Now let's say we have two torsion points $x_1,x_2$ that do not satisfy the inequality (\ref{eq:Elliptic1}). Thus we certainly have
$$ \log | \log_E \alpha - x_i|^{-1}_v > C (h_E(\alpha)+1) |F_i|^{\epsilon}$$
where $F_i$ is the Galois orbit of $x_i$. Assuming that $|F_1| \leq |F_2|$. by triangle inequality we have that
$$|x_1 - x_2| \leq |x_1 - \log_E \alpha| + |\log_E \alpha - x_2| \leq 2e^{-C (h_E(\alpha)+1)|F_1|^{\epsilon}}.$$
It follows that if $x_1 = b_{1,1} \omega_1 + b_{2,1} \omega_2$ and $x_2 = b_{1,2} \omega_1 + b_{2,2}\omega_2$, then the lowest common multiple of the denominators of $b_{i,j}$ have to be at least $O(e^{-C h_E(\alpha) |F_1|})$. In particular as $|F_2| \geq |F_1|$ we must have $\max \{ h(b_{1,2}), h(b_{2,2}) \} \geq \frac{C}{4} h_E(\alpha) |F_1|$. Increasing our constant $C$, we can guarantee that
$$|F_2| \geq \frac{1}{\frac{C}{4}h_E(\alpha)} e^{\frac{C}{4} h_E(\alpha)} \geq h_E(\alpha)^{2/\epsilon}.$$
This contradicts the upper bound in (\ref{eq:Elliptic2}). Thus for a given $\beta$, we can find a constant $C > 0$ such that 
$$\max_{x \in F} \log |x-\beta|^{-1}_v < C D^6 (\log D)^2 (h(\beta)+1)^2 |F|^{\epsilon},$$
is true for all Galois orbit $F$ of preperiodic points, with the possibility of one exception, as desired.
\end{proof}
 
We now handle the non-archimedean places. This will follow from a similar argument to the case of power maps, where we proved that no two roots of unity can be $p$-adically close to each other. First, we will state Cassel's theorem which will be essential.

\begin{theorem}[Theorem 3.4, \cite{Sil09}] \label{Cassels}
Let $K$ be an extension of $\mathbb{Q}_p$ with normalized valuation such that $|p|_v = \frac{1}{p}$ and let $E/K$ be an elliptic curve given by a Weierstrass equation 
$$y^2 + a_1xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6$$
with all $|a_i|_v \leq 1$. Let $P \in E(K)$ be a point of exact order $m \geq 2$. Then if $m$ is not a power of $P$, we have $|x(P)|_v,|y(P)|_v \leq 1$ and if $m = p^n$, we have 
$$|x(P)|_v, |y(P)|_v \leq \frac{3}{p^n - p^{n-1}}.$$ 
\end{theorem}


Let $v$ be a non-archimedean place. We first want to prove an analogue of Proposition \ref{DistanceBound} and Proposition \ref{DistanceBound2}, but there is no compactness available on $\mathbb{E}(\mathbb{C}_v)$. Instead, we will analyze directly the functions $\wp(z)$ and $\wp'(z)$. 
\par 
Passing to a finite extension, we may assume that $E$ has either good reduction at $v$ or is a Tate curve. If it is a Tate curve, then there exists $q \in \overline{K}_v$ with $|q|_v < 1$, such that $E$ can be given by the equation 
$$y^2 + xy = x^3 + a_4(q)x + a_6(q).$$
Here, $a_4(q), a_6(q)$ are power series in $q$ that converge. There is then a uniformization map, giving rise to an isomorphism $\overline{K_v}^*/q^{\mathbb{Z}} \to E(\overline{K})$ which is given by $u \mapsto (X(u,q),Y(u,q))$. The functions $X(u,q),Y(u,q)$ are power series in $u$ that converge everywhere. We may write $X(u,q)$ as 
$$X(u,q) = \frac{u}{(1-u)^2} + \sum_{n \geq 1} \left(\frac{q^n u}{(1-q^n u)^2} + \frac{q^{-n}u}{(1-q^{-n}u)^2} - 2 \frac{q^n}{(1-q^n)^2} \right)$$
and $Y(u,q)$ as
$$Y(u,q) = \frac{u^2}{(1-u)^3} + \sum_{n \geq 1} \left( \frac{(q^n u)^2 }{(1-q^n u)^3} + \frac{(q^{-n}u)^2}{(1-q^{-n}u)^3} + \frac{q^n}{1-q^n} \right).$$

The following proposition allows us to control the distances between points on $E(\mathbb{C}_v)$ in terms of distances of its images on $\mathbb{P}^1(\mathbb{C}_v)$ when $E$ is a Tate curve.

\begin{proposition} \label{NonArchimedeanDistanceBound1}
Assume that $z,w \in \mathbb{P}^1(\mathbb{C}_v)$ with $|z|,|w| \leq 1$ and $|z-w| < 1$. Then there exists $u,v \in \overline{K}_v$ with $|u| = |v| = 1$, $X(u,q) = z, X(v,q) = w$ and $|u-v| \leq |z-w|^{1/2}$. 
\end{proposition}


\begin{proof}
First, we may choose $u$ such that $X(u,q) = z$ and $q < |u|_v \leq 1$. Looking at the power series expansion of $X(u,q)$, we see that if $|u|_v < 1$, then $|z|_v > 1$ and so $|u|_v = 1$. Furthermore, $|1-u|_v = 1$ too. Now let $f(x) = X(x,q)$. Differentiating, we obtain 
$$f'(x) = -\frac{x+1}{(1-x)^3} + \sum_{n > 0} \frac{q^n (x q^n + 1)}{(1-q^n x)^3} + \sum_{n > 0} \frac{q^n (x+q^n)}{(q^n - x)^3}.$$
One can verify that all higher derivatives at $x = u$ have valuation $\leq 1$. Expanding out the Taylor series, we have
$$f(u+d) = f(u) + f'(u)d + \frac{f''(u)}{2} d^2 + \cdots$$
and as for a power series $h(z) = a_0 + a_1z + \cdots$, we must have $h(\overline{D}(0,r))$ containing $\overline{D}(a_0, \sup_{i} |a_i|_v r^i)$, it follows that $f(\overline{D}(u,r))$ contains $\overline{D}(z, |f'(u)|_v r)$ and $\overline{D}(z, |\frac{f''(u)}{2}|_v r^2)$ for $r \leq 1$.
\par 
If $|1+u|_v = 1$, then $|f'(u)|_v = 1$ and so $\overline{D}(z,r) \subseteq f(\overline{D}(u,r))$. In particular, we can choose some $v \in \overline{D}(u,|z-w|)$ that is the preimage of $w$ as $w \in \overline{D}(z,|z-w|)$. Then $|u-v| \leq |z-w|$. 
\par 
On the other hand if $|1+u|_v = 1$, we compute the second derivative to be 
$$f''(x) = -\frac{2(2+x)}{(1-x)^4} + O(q)$$
and since $|1+u|_v < 1$, we must have $|2+u|_v = 1$. Thus $|f''(u)/2|_v = 1$ and we get $\overline{D}(z,r^2) \subseteq f(\overline{D}(u,r))$. In this case, we can choose some $v \in \overline{D}(u,|z-w|^{1/2})$ that is the preimage of $w$ and so $|u-v| \leq |z-w|^{1/2}$. 
\end{proof}

We now do the case where $Y(u,q)$ is our projection map from $E \to \mathbb{P}^1$. 

\begin{proposition} \label{NonArchimedeanDistanceBound2}
Assume that $z,w \in \mathbb{P}^1(\mathbb{C}_v)$ with $|z|,|w| \leq 1$ and $|z-w| < 1$. Then there exists $u,v \in \overline{K}_v$ with $|u| = |v| = 1$, $Y(u,q) = z, Y(v,q) = w$ and $|u-v| \leq |z-w|^{1/3}$. 
\end{proposition}


\begin{proof}
The same argument in Proposition \ref{NonArchimedeanDistanceBound1} works, with a few tweaks. Again, we first note that we may choose $u$ such that $Y(u,q) = z$ and $q < |u|_v \le q1$. Then $|u|_v = |1-u|_v = 1$ too. Differentiating, we get
$$f'(x) = \frac{x(x+2)}{(1-x)^4} + \sum_{n > 0} \frac{q^n x (q^n x + 2)}{(1-q^n x)^4} + \sum_{n > 0} \frac{q^{2n} x( x + 2 q^n)}{(q^n - x)^4}.$$
We also compute the second derivative to be
$$f''(x) = \frac{2(x^2 + 4x + 1)}{(1-x)^5} + O(q)$$
and 
$$f'''(x) = \frac{6(x^2 + 6x + 3)}{(1-x)^6} + O(q).$$
If $|f'(u)|_v \not = 1$, then $|u+2|_v < 1$. Similarly if $|f''(u)/2|_v < 1$, then $|u^2 + 4u + 1|_v < 1$ and if $|f'''(u)/6|_v < 1$, then $|u^2 + 6u + 3|_v < 1$. We can then verify that at least one of $|f'(u)|_v, |f''(u)/2|_v$ and $|f'''(u)/6|_v$ must be equal to $1$. Thus $\overline{D}(u,r^3) \subseteq f(\overline{D}(u,r))$ for any $r < 1$, implying that we can choose $v$ with $Y(v,q) = w$ and $|u-v| \leq |z-w|^{1/3}$ as desired.
\end{proof}


We now are able to prove the analogue of Proposition \ref{PowerBound2} and \ref{PowerBound3}. Let's first make the following assumptions about our Lattès map. We first assume that our elliptic curve $E$ has either good reduction or is a Tate curve. In the good reduction case, let $y^2 = x^3 + ax + b$ be a minimal Weierstrass equation for our elliptic curve $E$. We then assume that our Lattès map arise from either the $x$-coordinate or $y$-coordinate map. In the case of a Tate curve, let $y^2 + xy = x^3 + a_4 x + a_6$ be our equation. Then rewriting 
$$x' = x - 1/12 \text{ and } y' = x/2,$$
we obtain a Weierstrass equation $y'^2 = x'^3 + ax + b$. We then again assume that our Lattès map arise from the $x$ or $y$-coordinate map in this case too.




\begin{proposition} \label{EllipticBound2}
Let $\varphi$ be a Lattès map for an elliptic curve $E$ defined over a number field $K$ and let $v$ be a non-archimedean place for $K$. Assume that $\varphi$ satisfy our above assumptions. Let $\beta \in \mathbb{P}^1(L)$ be non-preperiodic where $L$ is a number field with $[L:\mathbb{Q}] \leq D$. Then there exists at most one preperiodic point $x$ of degree $> p^2 D^2$ such that
$$\log|x-\beta|_v^{-1} > 3 \log p$$
where $p$ is the prime number corresponding to the place $v$.
\end{proposition}

\begin{proof}
Let's first handle the case where $E$ has good reduction at $v$. For any integer $n$, the multiplication by $n$ map, $[n]$, extends to the residue field too. In particular if $\varphi_n(z)$ is the rational map associated to it, then it has good reduction and so $\lambda_v(\varphi_n(x),\varphi_n(y)) \leq \lambda_v(x,y)$. Now let $x,y$ be preperiodic points of our Lattès map $\varphi(z)$ with $\lambda_v(x,y) < 1$. If $x$ corresponds to a torsion point $P$ on $E$ of order dividing $m$, which we may take to be even, it follows that $\varphi_{m}(x) = \infty$. Thus
$$\lambda_v(\infty, \varphi_{m}(y)) = \lambda_v(\varphi_{m}(x),\varphi_{m}(y)) \leq \lambda_v(x,y) < 1.$$
Thus $\varphi_{m}(y)$ is a rational number with $|\varphi_{m}(y)| > 1$. By Theorem \ref{Cassels}, it follows that if $Q$ is the torsion point corresponding to $y$ then $[m]Q$ has order a power of $p$ and 
$$|x([m]Q)|_v = |\varphi_m(y)|_v < p^{\frac{1}{p-1}} \implies \lambda_v(\infty, \varphi_m(y)) \geq p^{-\frac{1}{p-1}}.$$
 Thus we have
 $$\lambda_v(x,y) \geq p^{-\frac{1}{p-1}}$$
 for all preperiodic points $x,y$ and in particular, we cannot have $\log |x-\beta|_v^{-1} > \log p$ for two distinct preperiodic points. 
\par 
Now let's assume that $E$ is a Tate curve, given by the equation $y^2 + xy = x^3 + a_4x + a_6$. By our assumptions, our Lattès map arise from $x'$ or $y'$ where
$$x' = x - 1/12 \text{ and } y' = x/2.$$
Then for either $x'$ or $y'$, Proposition \ref{NonArchimedeanDistanceBound1} and \ref{NonArchimedeanDistanceBound2} still holds as translations do not matter, and dividing by a natural number only makes the image of our discs larger. 
\par 
Let $P,Q$ be two torsion points such that 
$$|\beta - \pi(P)|_v, |\beta - \pi(Q)|_v < \frac{1}{p^3}$$
where $\pi: E \to \mathbb{P}^1$ is either $x'$ or $y'$. Then $|\pi(P) - \pi(Q)| < \frac{1}{p^3}$ and we can then apply Proposition \ref{NonArchimedeanDistanceBound1} and \ref{NonArchimedeanDistanceBound2} to conclude the existence of $u,v$ corresponding to $P$ and $Q$ under $\pi$ such that $|u-v|_v \leq \frac{1}{p}$. Then as $|u|_v= |v|_v = 1$, we get that $|1 - \frac{v}{u}|_v < \frac{1}{p}$. As the torsion point $Q-P$ corresponds to $\frac{v}{u}$, we get that $|\pi(Q-P)|_v \geq p$. Cassel's theorem then says that we must have $P = Q$ and thus there is at most one preperiodic point $x$ with degree $> p^2 D^2$ such that $\log|x-\beta|^{-1}_v > 3 \log p$ as desired.
\end{proof}

As a similar method obtains for us the following analogue of Proposition \ref{PowerBound2}. 

\begin{proposition} \label{EllipticBound3}
Let $\varphi$ be a Lattès map for an elliptic curve $E$ defined over a number field $K$ and let $v$ be a non-archimedean place for $K$. Assume that $\varphi$ satisfy the assumptions as in Proposition \ref{EllipticBound2}. Let $D$ be a positive integer and $\delta > 0$. Then there are only finitely many preperiodic points $x$ such that 
$$1-\delta > |x-\beta|_v$$
for some non-preperiodic $\beta$ with $[K(\beta):K] \leq D$. 
\end{proposition}

\begin{proof}
First, if $|\beta|_v > 1$, then there is some $\delta$ such that $|\beta|_v \geq 1 + \delta$ as $\beta$ is defined over a degree $D$ field. Then by Cassel's theorem, there are only finitely many preperiodic $x$ with $|x| \geq 1+\delta$ and so we are done. 
\par 
Now if $|\beta|_v \leq 1$, we first choose a set of representative $\{\beta_1,\ldots,\beta_m\}$ such that they cover all residue classes of $\mathbb{A}^1(\mathbb{F}_q)$, for all $\mathbb{F}_q$ which are degree at most $D$ over the residue field of $K_v$. Then let $\{x_1,\ldots,x_{m}\}$ be a set of preperiodic points for $\varphi$ where $x_i$ is the same residue class as $\beta_i$. If there exists no such preperiodic point for $\beta_i$, we set $x_i = \beta_i$. 
\par 
Now let's assume for some $\beta \in \mathbb{P}^1(L)$ with $[L:K] \leq D$, we have a preperiodic point $x$ with $1 -\delta > |x-\beta|_v  $. Then we can find some $\beta_i,x_i$ such that $1 > |x-\beta_i|_v, |x-x_i|_v$. As $x_i$ lives in some number field of bounded degree, taking $\delta$ small enough we obtain $1-\delta > |x_i - \beta_i|_v$ for all $i$ and so $1-\delta > |x-x_i|_v$. If we have good reduction, similar to Proposition \ref{EllipticBound2}, if one lets $n_i$ be the order of $x_i$, we can apply $[n_i]$ to obtain 
$$\frac{1}{1-\delta} < |[n_i]x|_v.$$ 
In this case, there are only finitely many $[n_i]x$'s thanks to Cassel's theorem and so finitely many such $x$ in the first place.
\par 
Now if $E$ is a Tate curve, we can repeat the argument in Proposition \ref{EllipticBound2}. Again, $E$ is given by $y^2 + xy = x^3 + a_4x + a_6$, our projection maps are now given by 
$$x' = x - 1/12 \text{ or } y' = y - x/2.$$
Now, as $x'$ is just a translation of $x$, Proposition \ref{NonArchimedeanDistanceBound1} applies. For $y'$, both propositions certainly still hold as $x/2$ only makes the image of our discs larger. 
\par 
Thus we can find $u,v \in K^{*}/q^{\mathbb{Z}}$ such that $\pi(u) = x, \pi(v) = x_i$ and $|u-v| \leq (1-\delta')^3$ where $\pi: E \to \mathbb{P}^1$ is our projection map. Thus $|\frac{1}{ - \frac{u}{v}}|_v \leq \frac{1}{(1-\delta)^{1/3}}$. This implies that we can find points $P,Q$ on our elliptic curve corresponding to $x$ and $x_i$, satisfying $|\pi(P-P_i)| \geq \frac{1}{(1-\delta)^{1/3}}$. Again by Cassel's theorem, this tells us that there are only finitely many choices for $P-P_i$ and so we get finitely many preperiodic points $x$ that can be close to $\beta$ as desired.
\end{proof}

\subsection{Proof of Main Theorems}

We now prove Theorem \ref{IntroUniformIh4}. It will then be easy to see from the proof that we can similarly deduce Theorem \ref{IntroUniformIh1}, \ref{IntroUniformIh3} and \ref{IntroUniformIh2}. 

\begin{theorem} \label{UniformIh4}
Let $K$ be a number field and $S$ be a finite set of places of $K$. Let $\varphi: \mathbb{P}^1 \to \mathbb{P}^1$ be a Lattès map defined over $K$. Then there exists a constant $c > 0$, such that for all non-preperiodic $\beta \in \mathbb{P}^1(\overline{K})$, the set
$$\{x \in \operatorname{Prep}(\varphi) : \deg_{K}(x) > c \deg_{K}(\beta)^{20} \text{ and } x \text{ is S-integral relative to } \beta\}$$
is a union of at most $|S|$ $\operatorname{Gal}(\overline{K}/K)$-orbits.
\end{theorem}


\begin{proof}
Fix an $\epsilon > 0$. Let $\beta \in \mathbb{P}^1(L)$ where $L/K$ is an extension of degree at most $D$ and let $S_L$ be the set of places of $L$ above $K$. Let $M = \max \{ p^2 \mid v \text{ is a place above } p\}$. We first pass $K$ to a finite extension such that for each non-archimedean place $v \in S$, the elliptic curve $E$ associated to $\varphi$ has either good reduction or is a Tate curve. We wish to give an upper bound on $\log |x-\beta|^{-1}_w$ using Proposition \ref{EllipticBound1} and \ref{EllipticBound2}, but there are some assumptions on $\varphi$ for Proposition \ref{EllipticBound2}. However, we can modify $\varphi$ by a fixed constant $\alpha \in K$ such that $\alpha \varphi$ satisfies the assumptions required for each non-archimedean place $v \in S$, and this changes our upper bound on $\log |x-\beta|^{-1}_w$ by a constant only.
\par 
Thus applying Proposition \ref{EllipticBound1} and Proposition \ref{EllipticBound2}, we can find a constant $A_{\epsilon}$ such that for any place $w \in S_L$, we have
$$\max_{x \in F} \log |x-\beta|^{-1}_w < A_{\epsilon} D^6 (\log D)^2 (h(\beta)+1 + \log^+|\beta|_w)|F|^{\epsilon}$$
for all $\operatorname{Gal}(\overline{K}/K)$-orbits $F$ of preperiodic points with size $> MD^2$, with the possible exception of one orbit for each place $w$. But if $w$ and $w'$ live above the same place $v$, they differ by an automorphism of $\operatorname{Gal}(\overline{K}/K)$, and so we still pick up the same $\operatorname{Gal}(\overline{K}/K)$-orbit $F$ as an exception. Thus we get a total of $|S|$ many exceptions 
\par 
Now let $x$ be a preperiodic point with $\operatorname{Gal}(\overline{K}/K)$-orbit $F$, with $|F| > MD^2$ and is not one of the exceptions. Then Lemma \ref{GeneralQuantLog2} implies that there is a constant $C>0$ such that for each $w \in S_L$, we have
$$\left|\frac{1}{|F|} \sum_{x \in F} \lambda_w(x,\beta) - \int \lambda_w(x,\beta) d \mu_{\varphi,w} \right| \leq \frac{C A_{\epsilon}}{|F|^{1/2 - \epsilon}} \sqrt{\log |F|}  D^6 (\log D)^2 (h(\beta)+1 + \log^+|\beta|_w).$$
For any constant $N > 0$, assuming that $|F| > cD^{20}$ for some suitable constant $c$ gives us 
$$\left|\frac{1}{|F|} \sum_{x \in F} \lambda_w(x,\beta) - \int \lambda_w(x,\beta) \right| \leq \frac{h(\beta)+1 + \log^+|\beta|_w}{N D^{3.5}}.$$
Summing up over all places in $S_L$, as $\sum_{w \in S_L} N_w = \sum_{v \in S} N_v$ and also as $\sum_{w \in M_L} N_w \log^+|\beta|_w = h(\beta)$, by increasing our constant $c$ we get that
$$\left|\frac{1}{|F|} \sum_{w \in S_L} \sum_{x \in F} N_w \lambda_w(x,\beta) - \sum_{w \in S_L} N_w \int \lambda_w(x,\beta) d \mu_{\varphi,w} \right| $$
$$\leq \frac{h(\beta)+1}{ND^{3.5}}.$$
On the other hand, Proposition \ref{AraZhEquib2} tells us that if $c$ is large enough, then
$$|h_{\overline{L}_{\beta}}(F) - \langle \overline{L}_{\beta}, \overline{L}_{\varphi} \rangle| \leq \frac{1}{D^{3.5}}.$$
Expanding out, we get
$$\frac{1}{|F|} \sum_{x \in F} \sum_{w \in M_K} N_w \lambda_w(x,\beta) - \sum_{w \in M_L} N_w \int \lambda_w(x,\beta) d \mu_{\varphi,w} \geq h_{\varphi}(\beta) - \frac{1}{D^{3.5}}.$$
Now if $\beta$ were $S$-integral relative to $x$, we would get that $\lambda_w(x,\beta) = 0$ for all $w \not \in S_L$. Using our bound for $\frac{1}{|F|}\sum \lambda_w(x,\beta) - \int \lambda_w(x,\beta)$, we get
$$h_{\varphi}(\beta) \leq \frac{h(\beta)+1}{ND^{3.5}} + \frac{1}{D^{3.5}}.$$
Now we know that $h_{\varphi}(\beta)$ and $h(\beta)$ differs by at most a constant depending on $\varphi$, and so if $h_{\varphi}(\beta)$ were larger than some constant, we would get a contradiction. On the other hand since $\beta$ lives over a degree $D$ field, by \cite{Mas89} we know that $h_{\varphi}(\beta) \geq \frac{c}{D^3 (\log D)^2}$ and in particular we get a contradiction for $N$ large enough if the height of $h_{\varphi}(\beta)$ was small too. 
\end{proof}


Adapting the argument and using Proposition \ref{PowerBound1} and Corollary \ref{PowerBound3}, we obtain Theorem \ref{IntroUniformIh3}.

\begin{theorem} \label{UniformIh3}
Let $S$ a finite set of places of $\mathbb{Q}$. Then there exists a constant $c > 0$, such that for all $\beta \in \overline{\mathbb{Q}}^{\times} \setminus \mu_{\infty}$, the set
$$\{\zeta \in \mu_{\infty} : \deg_{\mathbb{Q}}(\zeta) > c \deg_{\mathbb{Q}}(\beta)^{10} \text{ and } \zeta \text{ is S-integral relative to } \beta\}$$
is a union of at most $|S_{\operatorname{fin}}|$ $\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-orbits.
\end{theorem}

\begin{proof}
Here, the same argument works to give us that except for $|S_{\operatorname{fin}}|$ Galois orbits, we have 
$$\left|\frac{1}{|F|} \sum_{w \in S_L} \sum_{x \in F} N_w \lambda_w(x,\beta) - \sum_{w \in S_L} N_w \int \lambda_w(x,\beta) d \mu_{\varphi,w} \right| \leq \frac{h(\beta)+1}{ND^{1.5}}.$$
Proposition \ref{AraZhEquib2} again tells us that 
$$|h_{\overline{L}_{\beta}}(F) - \langle \overline{L}_{\beta}, \overline{L}_{\varphi} \rangle| \leq \frac{1}{D^{1.5}}.$$
If $\beta$ were then $S$-integral relative to $x$, we would get 
$$h_{\varphi}(\beta) = h(\beta) \leq \frac{h(\beta)+1}{ND^{1.5}} + \frac{1}{D^{1.5}}.$$
If $h(\beta) \geq 1$, then we get a contradiction. Else a result of Dobrowolski's \cite{Dob79} implies that
$$h(\beta) \geq \frac{c}{D (\log D)^3}$$
and so we get a contradiction for $N$ large enough. 
\end{proof}

Using Proposition \ref{PowerBound2} instead of Corollary \ref{PowerBound3} allows us to have no exception, although we do not have a nice control in terms fo the degree. This gives us Theorem \ref{IntroUniformIh1}.

\begin{theorem} \label{UniformIh1}
Let $S$ be a finite set of places of $\mathbb{Q}$ and $K$ a number field. Then there exists a constant $C = C([K:\mathbb{Q}],S)$ such that the following holds: for any $\beta \in K^* \setminus \mu_{\infty}$, if $\zeta \in \mu_{\infty}$ is $S$-integral relative to $\beta$ then $\deg_{\mathbb{Q}}(x) < C$.
\end{theorem} 

Finally, using Proposition \ref{EllipticCMBound1} and \ref{EllipticBound3} gives us Theorem \ref{IntroUniformIh2}.

\begin{theorem} \label{UniformIh2}
Let $K$ be a number field and $S$ be a finite set of places of $K$. Let $\varphi: \mathbb{P}^1 \to \mathbb{P}^1$ be a Lattès map defined over $K$ that is associated to an elliptic curve $E$ with complex multiplication and let $L$ be a finite extension. Then there exists a constant $C = C([L:K],S)$ such that the following holds: for any $\beta \in \mathbb{P}^1(L) \setminus \operatorname{Prep}(\varphi)$, if $x \in \operatorname{Prep}(\varphi)$ is $S$-integral relative to $\beta$ then $\deg_{K}(x) < C$.
\end{theorem}

