On Szemerédi's theorem with differences from a random set

Abstract

We consider, over both the integers and finite fields, Szemerédi's theorem on $k$-term arithmetic progressions where the set $S$ of allowed common differences in those progressions is restricted and random. Fleshing out a line of enquiry suggested by Frantzikinakis et al, we show that over the integers, the conjectured threshold for $\mathbb{P}(d\in S)$ for Szemerédi's theorem to hold a.a.s follows from a conjecture about how so-called dual functions are approximated by nilsequences. We also show that the threshold over finite fields is different to this threshold over the integers.

14 pages, minor changes from previous version