Boundedness of $\gcd(|x-y|,|a_x-a_y|)$ in sequence

Question

Let $a_1,a_2,\ldots$ be an infinite sequence of distinct positive integers, and let $n$ be a positive integer. Does there always exist integers $x,y$ such that $\gcd(|x-y|,|a_x-a_y|)>n$?

When $a_x=x+c$ for some fixed $c$, the statement is true because $\gcd(|x-y|,|x-y|)=|x-y|$, which is unbounded.

Answer 1

By pigeonhole principle, there exist different natural numbers $x$ and $y$ such that $x\equiv y \pmod {n+1}$ and $a_x\equiv a_y \pmod {n+1}$. Then both $|x-y|$ and $|a_x-a_y|$ are divisible by $n+1$.