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Published March 2020 | Submitted
Journal Article Open

Improved Bounds for Progression-Free Sets in C^n₈

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(C₄^n) ≤ (3.611)^n, where C_m denotes the cyclic group of order m. For finite abelian groups G≅∏^n_(i=1), where m₁,…,m_n denote positive integers such that m₁ |…|m_n, this also yields a bound of the form r₃(G)⩽(0.903)^(rk₄(G))|G|, with rk₄(G) representing the number of indices i ∈ {1,…, n} with 4 |m_i. In particular, r₃(Cn₈) ≤ (7.222)^n. In this paper, we provide an exponential improvement for this bound, namely r₃(Cn₈) ≤ (7.0899)^n.

Additional Information

© 2020 The Hebrew University of Jerusalem. First Online: 12 February 2020. Research supported by Russian Science Foundation grant 17-71-20153.

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