Let $\delta >0$ and $k\ge 3$. Is it true that there exists $m\ge 1$ (depending only on $\delta$ and $k$) such that, for all large $N$, if $A,B\subseteq \{1,\dots ,N\}$ with $|A|\ge \delta N$ and $|B|\ge m$ then there is a non-trivial $k$-term arithmetic progression in $A$ whose common difference is in $B-B$?