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 \(2n-3\).
Reviewer: I.Z.Ruzsa (Budapest)
MSC:
20K01 | Finite abelian groups |
11B75 | Other combinatorial number theory |
20D60 | Arithmetic and combinatorial problems involving abstract finite groups |