Let $A \subset \mathbb{Z}/N\mathbb{Z}$, let $s \mid N$ with $s + 1 < \min\{|A|,\sqrt{N}\}$ and let $B = \{ ks \ : \ 0 \leq k \leq s\}$. Denote $$ E(A, B) = \#\{ (a,a',b,b') \in A^2 \times B^2 \ : \ a' - a \equiv b - b' \pmod{N}\}. $$ If $E(A, B) \geq \delta |A||B|^2$ for some $\delta \in (0,1)$, what can we say about $A$?
I think that if $E(A,B)$ is large, then, as $B$ is an arithmetic progression with common difference $s$, $A$ should contain lots of arithmetic progressions with common difference $s$ as well. However I wonder what a more precise quantitative statement one could have here about $A$ in terms of $\delta$? Thanks a lot.