Erdős Problem #1185 — Dense progressions with differences from a prescribed difference set

For every $δ>0$ and integer $k≥3$, is there an $m=m(δ,k)$ such that, for all sufficiently large $N$, whenever $A,B⊆{1,…,N}$ satisfy $|A|≥δN$ and $|B|≥m$, the set $A$ contains a nonconstant $k$-term arithmetic progression whose common difference lies in $B-B$?