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

TL;DR

This work establishes a quantitative bound on the size of restricted 3-term arithmetic progression free sets in the finite field model $\mathbb{F}_p^n$ when the common difference lies in $\{0,1,2\}^n$. The authors develop an inverse-type stability theorem inspired by CSPs, and convert correlations into density increments via random restrictions and specialized basis changes, enabling an iterative increment argument. The main result shows that the maximum density decays as $\mu(A) \le \frac{C}{(\log\log\log n)^c}$ for some $c,C>0$ depending only on $p$, which improves the previous bound of $O(1/\log^{*} n)$ and advances understanding of restricted-pattern phenomena in the finite field setting. The techniques—CSP stability, Efron-Stein decompositions, random restrictions, and carefully designed basis changes—offer tools potentially applicable to broader constrained-pattern problems and CSP approximability in additive combinatorics.

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.

Theorems & Definitions (31)

• Theorem 1.1
• Remark 1.2
• Definition 1.3
• Definition 1.4
• Definition 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5