  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 $r_{3}(C_{8}^{n}) \leq (7.09)^{n}$.
