k-Sums in abelian groups

TL;DR

This paper investigates the size of sumsets formed by k distinct elements in finite subsets of abelian groups, establishing lower bounds and characterizations for when these sumsets are minimal, thus answering a question posed by Diderrich.

Contribution

It provides a lower bound for the size of k-sumsets in abelian groups and characterizes subsets where the sumset size equals the original set size for some k, extending previous work.

Findings

|k ∧ A| >= |A| for most k, with specific exceptions

Characterization of subsets with minimal sumsets

Answer to Diderrich's question

Abstract

Given a finite subset A of an abelian group G, we study the set k \wedge A of all sums of k distinct elements of A. In this paper, we prove that |k \wedge A| >= |A| for all k in {2,...,|A|-2}, unless k is in {2,|A|-2} and A is a coset of an elementary 2-subgroup of G. Furthermore, we characterize those finite subsets A of G for which |k \wedge A| = |A| for some k in {2,...,|A|-2}. This result answers a question of Diderrich. Our proof relies on an elementary property of proper edge-colourings of the complete graph.