HAL: hal-00630441, version 2

arXiv: 1110.1961

DOI: 10.1017/S0963548312000168

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Combinatorics, Probability and Computing 21, 4 (2012) 582-596

Available versions:

v1 (2011-10-10)

v2 (2012-06-26)

k-sums in abelian groups

Benjamin Girard (  Corresponding author ) 1, Simon Griffiths 2, Yahya Ould Hamidoune 1

(2012)

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.

1:	Institut de Mathématiques de Jussieu (IMJ)
	CNRS : UMR7586 – Université Pierre et Marie Curie [UPMC] - Paris VI – Université Paris VII - Paris Diderot
2:	Instituto Nacional de Matemática Pura e Aplicada (IMPA)
	Instituto Nacional de matematica pura e aplicada

Subject

:

Mathematics/Combinatorics Mathematics/Number Theory Mathematics/Group Theory

Keyword(s): Additive combinatorics - restricted set addition - abelian groups - proper edge-colourings

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From: Benjamin Girard <>

Submitted on: Tuesday, 26 June 2012 16:26:04

Updated on: Tuesday, 26 June 2012 20:50:12