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On restricted sums. (English) Zbl 0974.20043

Let \(A\) be a finite set in a commutative group, \(|A|=n\). It is proved that for \(n\geq 33\), the number of elements representable in the form \(a+a'\), with \(a,a'\in A\) and \(a\neq a'\), is at least \(3n/2\) except when \(A\) is contained in a subgroup of \(<3n/2\) elements. Equality occurs when \(A\) is the union of two cosets of a subgroup. This complements a result of Dias da Silva and Hamidoune, which asserts that in case of a cyclic group of prime order the corresponding cardinality is at least \(2n-3\).

MSC:

20K01 Finite abelian groups
11B75 Other combinatorial number theory
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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