Determine whether for every real number $t$ such that $0<t<\frac{1}{2}$ there exists an infinite set $S$ of positive integers such that  
$$  
|x-m y|>t y  
$$  
for every pair of different elements $x$ and $y$ of $S$ and every positive integer $m$ (i.e. $m>0$ ).The original answer is in the form of $t<\left\{\frac{s_{n-1}}{s_{n}}\right\}<\frac{a}{b}$. Find the value of $a+b$.

Let's think step by step and output the final answer within \boxed{}.
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