Improved Bounds for Progression-Free Sets in C₈ⁿ

2018/05/15 by Petrov, Fedor, Pohoata, Cosmin
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1805.05549

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(C₄ⁿ) \leqslant (3.61)ⁿ, where C_m denotes the cyclic group of order m. For finite abelian groups G ≅ ∏_i=1ⁿ C_m_i, where m₁,…,m_n denote positive integers such that m₁ | … | m_n, this also yields a bound of the form r₃(G) \leqslant (0.903)^rk₄(G) |G|, with rk₄(G) representing the number of indices i ∈ \1,…,n\ with 4 | m_i. In particular, r₃(C₈ⁿ) \leqslant (7.22)ⁿ. In this paper, we provide an exponential improvement for this bound, namely r₃(C₈ⁿ) ≤ (7.09)ⁿ.

Related

• Complete 3-term arithmetic progression free sets of small size in vector spaces and other abelian groups
• Caps and progression-free sets in ℤ_mⁿ
• The Erdős-Ginzburg-Ziv constant and progression-free subsets
• New lower bounds for three-term progression free sets in \mathbbF_pⁿ
• A generalization of Croot-Lev-Pach's Lemma and a new upper bound for the size of difference sets in polynomial rings