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On Sets with a Small Subset Sum

Published online by Cambridge University Press:  01 September 1999

Y. O. HAMIDOUNE
Affiliation:
E. Combinatoire, Univ. Pierre et Marie Curie, 4 Place Jussieu, 75230 Paris Cedex 05, France (e-mail: yha@ccr.jussieu.fr)
A. S. LLADÓ
Affiliation:
Dep. Matemàtica Aplicada i Telemàtica, Univ. Politècnica de Catalunya, Jordi Girona, 1, 08034 Barcelona, Spain (e-mail: allado@mat.upc.es, oriol@mat.upc.es)
O. SERRA
Affiliation:
Dep. Matemàtica Aplicada i Telemàtica, Univ. Politècnica de Catalunya, Jordi Girona, 1, 08034 Barcelona, Spain (e-mail: allado@mat.upc.es, oriol@mat.upc.es)

Abstract

Let A be a subset of an abelian group G. The subset sum of A is the set [sum ](A) = {[sum ]xT[mid ]TA}. We prove the following result. Let S be a generating subset of an abelian group G such that 0∉S and 14[les ][mid ]S[mid ]. Then one of the following conditions holds.

(i) [mid ][sum ](S)[mid ][ges ]min([mid ]G[mid ] −3, 3[mid ]S[mid ]−3).

(ii) There is an xS such that S[setmn ]{x} generates a proper subgroup of order less than (3[mid ]S[mid ]−3)/2.

As a consequence, we obtain the following open case of an old conjecture of Diderrich. Let q be a composite odd number and let G be an abelian group of order 3q. Let S be a subset of G with cardinality q+1. Then every element of G is the sum of some subset of S.

Type
Research Article
Copyright
1999 Cambridge University Press

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