Popular progression differences in vector spaces

TL;DR

This paper improves bounds on the density of three-term arithmetic progressions in subsets of vector spaces over finite fields, demonstrating the necessity of tower-type bounds from regularity lemmas.

Contribution

It establishes tight bounds on the dimension growth needed for arithmetic progression density results, showing tower-type bounds are essential.

Findings

Proves the density of 3-term progressions is at least the random bound in large finite field vector spaces.

Shows the dimension must grow as an exponential tower of height proportional to log(1/epsilon).

First example where regularity lemma bounds are proven necessary for such results.

Abstract

Green proved an arithmetic analogue of Szemer\'edi's celebrated regularity lemma and used it to verify a conjecture of Bergelson, Host, and Kra which sharpens Roth's theorem on three-term arithmetic progressions in dense sets. It shows that for every subset of $\mathbb{F}_p^n$ with $n$ sufficiently large, the density of three-term arithmetic progressions with some nonzero common difference is at least the random bound (the cube of the set density) up to an additive $\epsilon$. For a fixed odd prime $p$, we prove that the required dimension grows as an exponential tower of $p$'s of height $\Theta(\log(1/\epsilon))$. This improves both the lower and upper bound, and is the first example of a result where a tower-type bound coming from applying a regularity lemma is shown to be necessary.