Improved Bounds for Progression-Free Sets in $C_{8}^{n}$

TL;DR

This paper improves the upper bounds on the size of the largest progression-free subsets in the group $C_{8}^{n}$, refining previous exponential bounds and advancing understanding of progression-free sets in finite abelian groups.

Contribution

It provides a tighter exponential bound for progression-free subsets in $C_{8}^{n}$, improving upon previous bounds by Croot, Lev, and Pach.

Findings

Bound for $r_3(C_8^n)$ improved to $(7.09)^n$

Enhanced understanding of progression-free sets in finite abelian groups

Refined exponential bounds for specific cyclic groups

Abstract

Let $G$ be a finite group, and let $r_{3}(G)$ represent the size of the largest subset of $G$ without non-trivial three-term progressions. In a recent breakthrough, Croot, Lev and Pach proved that $r_{3}(C_{4}^{n}) \leqslant (3.61)^{n}$, where $C_{m}$ denotes the cyclic group of order $m$. For finite abelian groups $G \cong \prod_{i=1}^{n} C_{m_{i}}$, where $m_{1},\ldots,m_{n}$ denote positive integers such that $m_{1} | \ldots | m_{n}$, this also yields a bound of the form $r_{3}(G) \leqslant (0.903)^{\operatorname{rk}_{4}(G)} |G|$, with $\operatorname{rk}_{4}(G)$ representing the number of indices $i \in \left\{1,\ldots,n\right\}$ with $4\ |\ m_{i}$. In particular, $r_{3}(C_{8}^{n}) \leqslant (7.22)^{n}$. In this paper, we provide an exponential improvement for this bound, namely $r_{3}(C_{8}^{n}) \leq (7.09)^{n}$.