Improved Bounds for Progression-Free Sets in $C_8^n$

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 $r_3(C_4^n)⩽(3.61{)}^n$, where $C_m$ denotes the cyclic group of order $m$. For finite abelian groups $G\cong {\prod }_{i=1}^n{C}_{m_i}$, where $m_1,\dots ,m_n$ denote positive integers such that $m_1|\dots |m_n$, this also yields a bound of the form $r_3(G)⩽(0.903{)}^{{\mathrm{rk}}_4(G)}|G|$, with ${\mathrm{rk}}_4(G)$ representing the number of indices $i\in \left\{1,\dots ,n\right\}$ with $4|m_i$. In particular, $r_3(C_8^n)⩽(7.22{)}^n$. In this paper, we provide an exponential improvement for this bound, namely $r_3(C_8^n)\le (7.09{)}^n$.

14 pages