Let $|\cdot |$ denote the cardinality of a set. Is it possible to find a sequence of finite subsets $(A_n)_{n=1}^{\infty}$ of $\mathbb N$ such that
$\lim\limits_n \min A_n=\infty$;
for every (fixed) $k\in \mathbb N$ we have $$\frac{|A_n\cap A_{n+k}|}{\sqrt{|A_n|\cdot |A_{n+k}|}}$$ goes to $1$ exponentially fast with respect to $n$?
For example, if $A_n=\{n,\ldots 2n-1\}$, the above conditions hold but with polynomial speed for (2).
Thanks!