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On Restricted Sums

Published online by Cambridge University Press:  09 April 2001

Y. O. HAMIDOUNE ,
A. S. LLADÓ  and
O. SERRA
Y. O. HAMIDOUNE
Affiliation:
UFR 921, E. Combinatoire, Université Pierre et Marie Curie, 4 Place Jussieu, 75005 Paris, France (e-mail: yha@ccr.jussieu.fr)
A. S. LLADÓ
Affiliation:
Dep. Matematica Aplicada i Telematica, Universitat Politècnica de Catalunya, Jordi Girona, 1, 08034 Barcelona, Spain (e-mail: allado@mat.upc.es, oserra@mat.upc.es)
O. SERRA
Affiliation:
Dep. Matematica Aplicada i Telematica, Universitat Politècnica de Catalunya, Jordi Girona, 1, 08034 Barcelona, Spain (e-mail: allado@mat.upc.es, oserra@mat.upc.es)
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Abstract

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Let G be an abelian group. For a subset A ⊂ G, denote by 2 ∧ A the set of sums of two different elements of A. A conjecture by Erdős and Heilbronn, first proved by Dias da Silva and Hamidoune, states that, when G has prime order, [mid ]2 ∧ A[mid ] [ges ] min([mid ]G[mid ], 2[mid ]A[mid ] − 3).

We prove that, for abelian groups of odd order (respectively, cyclic groups), the inequality [mid ]2 ∧ A[mid ] [ges ] min([mid ]G[mid ], 3[mid ]A[mid ]/2) holds when A is a generating set of G, 0 ∈ A and [mid ]A[mid ] [ges ] 21 (respectively, [mid ]A[mid ] [ges ] 33). The structure of the sets for which equality holds is also determined.

Information

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 9 , Issue 6 , November 2000 , pp. 513 - 518
Copyright
2000 Cambridge University Press