Effective Bounds for Restricted $3$-Arithmetic Progressions in $\mathbb{F}_p^n$

TL;DR

This paper establishes new upper bounds on the density of subsets in finite vector spaces over prime fields that avoid certain restricted 3-term arithmetic progressions, improving upon previous bounds derived from the density Hales-Jewett theorem.

Contribution

It provides the first non-trivial bounds for the density of sets avoiding restricted 3-term progressions in _p^n, using novel techniques beyond the density Hales-Jewett theorem.

Findings

Density of progression-free sets is at most C/(log log log n)^c

Improves previous bounds from O(1/log^* n)

First reasonable bounds for such sets

Abstract

For a prime $p$, a restricted arithmetic progression in $\mathbb{F}_p^n$ is a triplet of vectors $x, x+a, x+2a$ in which the common difference $a$ is a non-zero element from $\{0,1,2\}^n$. What is the size of the largest $A\subseteq \mathbb{F}_p^n$ that is free of restricted arithmetic progressions? We show that the density of any such a set is at most $\frac{C}{(\log\log\log n)^c}$, where $c,C>0$ depend only on $p$, giving the first reasonable bounds for the density of such sets. Previously, the best known bound was $O(1/\log^{*} n)$, which follows from the density Hales-Jewett theorem.