A Note on Lower Bounds in Szemer\'edi's Theorem with Random Differences

TL;DR

This paper improves lower bounds on the size of the difference set in Szemerédi's theorem over finite fields, using probabilistic and tensor rank techniques to establish when the theorem holds with high probability.

Contribution

It introduces a novel combination of existing methods to slightly enhance the known lower bounds for the difference set size in Szemerédi's theorem over finite fields.

Findings

Improved lower bounds for difference set size in Szemerédi's theorem.

Application of tensor rank bounds to additive combinatorics.

Asymptotic almost sure validity of the theorem under new bounds.

Abstract

In this note, we consider Szemer\'{e}di's theorem on $k$-term arithmetic progressions over finite fields $\mathbb{F}_p^n$, where the allowed set $S$ of common differences in these progressions is chosen randomly of fixed size. Combining a generalization of an argument of Altman with Moshkovitz--Zhu's bounds for the partition rank of a tensor in terms of its analytic rank, we (slightly) improve the best known lower bounds (due to Bri\"et) on the size $|S|$ required for Szemer\'edi's theorem with difference in $S$ to hold asymptotically almost surely.