k-Sums in abelian groups

Details k-Sums in abelian groups k-Sums in abelian groups

2011-10-10

arxiv.org/abs/1110.1961v1

Mathematics

Combinatorics

15 pages, submitted

Scientific paper

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 $|k \wedge A| \geq |A|$ for all $k \in {2,...,|A|-2}$, unless $k \in {2,|A|-2}$ and $A$ is a coset of an elementary 2-subgroup of $G$. Furthermore, we characterise those finite sets $A {\ss}q G$ 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.

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