On the Sumset of Sets of Size k

2025/05/12 by Vincent Schinina, Schinina, Vincent · 1 citation
Mathematics · Computer Science · #Limits and Structures in Graph Theory #Analytic Number Theory Research #Advanced Graph Theory Research

paper · pdf · doi:10.48550/arxiv.2505.07679

Abstract

The set R_G(h,k) consists of all possible sizes for the h-fold sumset of sets containing k elements from an additive abelian group G. The exact makeup of this set is still unknown, but there has been progress towards determining which integers are present. We know that R_G(h,k)⊆[hk-h+1,\binomh+k-1h], where the right side is an interval of integers that includes the endpoints. These endpoints are known to be attained. We will prove that the integers in [hk-h+2,hk-1] are not possible sizes for the h-fold sumset of a set containing k≥ 4 elements of a torsion-free additive abelian group G. Furthermore, we will confirm that this interval can't be made larger by exhibiting a subset of G whose h-fold sumset has size hk.

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