Improved bounds for progression-free sets in $$C_8^{n}$$C8n

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^n_4) ⩽ (3.61)^n, where C_m denotes the cyclic group of order m. For finite abelian groups G ≅ ∏^n_(i=1) C_(m_i), where m₁,…,m_n denote positive integers such that m₁|…|m_n, this also yields a bound of the form r_3(G) ⩽ (0.903)^(rk₄(G))|G|, with rk₄(G) representing the number of indices I ∈ {1,…,n} with 4 | m_i. In particular, r_3(C^n_8) ⩽ (7.22)^n. In this paper, we provide an exponential improvement for this bound, namely r_3(C^n_8) ≤ (7.09)^n.