The structure of sets with cube-avoiding sumsets

2024/11/21 by Karam, Thomas, Keevash, Peter
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2411.14145

Abstract

We prove that if d ≥ 2 is an integer, G is a finite abelian group, Z₀ is a subset of G not contained in any strict coset in G, and E₁,…,E_d are dense subsets of Gⁿ such that the sumset E₁+…+E_d avoids Z₀ⁿ then E₁, …, E_d essentially have bounded dimension. More precisely, they are almost entirely contained in sets E₁' × G^Ic, …, E_d' × G^Ic, where the size of I ⊂ [n] is non-zero and independent of n, and E₁',…,E_d' are subsets of G^I such that the sumset E₁'+…+E_d' avoids Z₀^I.

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