Sum-free sets in Z₅ⁿ

2023/01/17 by Vsevolod F. Lev, Lev, Vsevolod F.
Mathematics · Social Sciences · #Advanced Topology and Set Theory #FOS: Mathematics #Japanese History and Culture #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2301.06750

openalex publication_date 2023/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It is well-known that for a prime p≡ 2\pmod 3 and integer n≥ 1, the maximum possible size of a sum-free subset of the elementary abelian group \mathbb Z_pⁿ is \frac13 (p+1)p^n-1. We establish a matching stability result in the case p=5: if A⊆\mathbb Z₅ⁿ is a sum-free subset of size |A|>\frac32⋅5^n-1, then there are a subgroup H<\mathbb Z₅ⁿ of size |H|=5^n-1 and an element e∉ H such that A⊆(e+H)∪(-e+H).

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