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

Abstract

Let G be a finite group, and let r3(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 r3($$C_4^{n}$$) ≤ (3.611)n, where Cm denotes the cyclic group of order m. For finite abelian groups $$G \cong \prod\nolimits_{i = 1}^n {{C_{{m_i}}}} $$, where m1,…,mn denote positive integers such that m1 |…|mn, this also yields a bound of the form $$r_3(G)\leqslant(0.903)^{{rk}_4(G)}|G|$$, with rk4(G) representing the number of indices i ∈ {1,…, n} with 4 |mi. In particular, r3($$C_8^{n}$$) ≤ (7.222)n. In this paper, we provide an exponential improvement for this bound, namely r3($$C_8^{n}$$) ≤ (7.0899)n.