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

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$C_{m}$denotesthecyclicgroupoforder$m$. For finite abelian groups $G\cong {\prod }_{i=1}^n{C}_{m_i}$, where $m_1,\dots ,m_n$ denote positive integers such that $m_1\mathrm{\mid }\dots \mathrm{\mid }{m}_n$, this also yields a bound of the form $r_3(G)⩽(0.903{)}^{{\mathrm{rk}}_4(G)}\mathrm{\mid }G\mathrm{\mid }$, with ${\mathrm{rk}}_4(G)$ representing the number of indices $i \in \left{1,\ldots,n\right}$with$4\ |\ m_{i}$\mathrm{.}Inparticular,$r_{3}(C_{8}^{n}) \leqslant (7.22)^{n}$. In this paper, we provide an exponential improvement for this bound, namely $r_3(C_8^n)\le (7.09{)}^n$.