Popular progression differences in vector spaces II

TL;DR

This paper investigates the size of vector spaces needed to guarantee the existence of three-term arithmetic progressions with a certain density, extending previous results with precise tower height bounds for large primes.

Contribution

It determines the tower height of the minimal dimension for arithmetic progression density guarantees in vector spaces over finite fields, refining bounds for large primes.

Findings

For primes p ≥ 19, the tower height of n_p(α,β) is characterized up to a constant factor.

When β=α^3/2, the required dimension is a tower of twos with height proportional to (log p)(log log(1/α)).

Different regions of α and β require different bounds and arguments to estimate the tower height.

Abstract

Green used an arithmetic analogue of Szemer\'edi's celebrated regularity lemma to prove the following strengthening of Roth's theorem in vector spaces. For every $\alpha>0$, $\beta<\alpha^3$, and prime number $p$, there is a least positive integer $n_p(\alpha,\beta)$ such that if $n \geq n_p(\alpha,\beta)$, then for every subset of $\mathbb{F}_p^n$ of density at least $\alpha$ there is a nonzero $d$ for which the density of three-term arithmetic progressions with common difference $d$ is at least $\beta$. We determine for $p \geq 19$ the tower height of $n_p(\alpha,\beta)$ up to an absolute constant factor and an additive term depending only on $p$. In particular, if we want half the random bound (so $\beta=\alpha^3/2$), then the dimension $n$ required is a tower of twos of height $\Theta \left((\log p) \log \log (1/\alpha)\right)$. It turns out that the tower height in general takes on a different form in several different regions of $\alpha$ and $\beta$, and different arguments are used both in the upper and lower bounds to handle these cases.