Relative sizes of iterated sumsets

2024/12/24 by Kravitz, Noah · 6 citations
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2412.18598

Abstract

Let hA denote the h-fold sumset of a subset A of an abelian group. Resolving a problem of Nathanson, we show that for any prescribed permutations σ₁, …, σ_H ∈ \mathfrakS_n, there exist finite subsets A₁, …, A_n ⊆ ℤ such that for each 1 ≤ h ≤ H, the relative order of the quantities |h A₁|, …, |h A_n| is given by σ_h. We also establish extensions where ℤ is replaced by any other infinite abelian group or where one prescribes some equalities (not only inequalities) among the sumset sizes.

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