Adding distinct congruence classes. (English) Zbl 0894.11004
The main result of the paper asserts that for any finite abelian group \(G\) and generating subset \(S\) such that \(0\not\in S\) and \( | S | \geq 5\), the number of elements representable as a sum of certain elements of \(S\) is \(\geq \min ( | G | -1, 2 | S |)\). The same holds for noncommutative groups if rearrangement of the elements is permitted, and the \(-1\) can be removed for cyclic groups. These results are best possible. Similar estimates were obtained previously under further assumptions by J. E. Olson [Acta Arith. 28, 147-156 (1975; Zbl 0318.10035)] and E. T. White [J. Comb. Theory, Ser. A 24, 118-121 (1978; Zbl 0369.20021)].
Reviewer: I.Z.Ruzsa (Budapest)
MSC:
11B13 | Additive bases, including sumsets |
11B75 | Other combinatorial number theory |
20F05 | Generators, relations, and presentations of groups |
05E15 | Combinatorial aspects of groups and algebras (MSC2010) |
20K01 | Finite abelian groups |