Erdős Problem #1185

Let \(\delta>0\) and \(k\geq 3\). Is it true that there exists \(m\geq 1\) (depending only on \(\delta\) and \(k\)) such that, for all large \(N\), if \(A,B\subseteq \{1,\ldots,N\}\) with \(\lvert A\rvert \geq \delta N\) and \(\lvert B\rvert \geq m\) then there is a non-trivial \(k\)-term arithmetic progression in \(A\) whose common difference is in \(B-B\)? A problem of Erdős and Mauldin [Er80, p. 92], who ask the case \(k=3\) and say that the same probably holds for \(k\) terms. The answer is no, already for \(k=3\): to disprove it suffices to give an infinite set \(B\) such that \(B-B\) is not \(2\)-intersective (a set \(S\) is \(k\)-intersective if every subset of \(\mathbb{N}\) with positive density contains an arithmetic progression of length \(k+1\) whose common difference is in \(S\)), which was provided by Furstenberg [Fu81]. The statement below is the negation of the claim for all \(\delta>0\) and \(k\geq 3\), with \(m\) chosen before the threshold \(N_0\) for "large \(N\)".