Restricted set addition in groups. II: A generalization of the Erdős-Heilbronn conjecture. (English) Zbl 0973.11026

Lev, Vsevolod F.

Electron. J. Comb. 7, No. 1, Research paper R4, 10 p. (2000); printed version J. Comb. 7, No. 1 (2000).

[For Part I, see J. Lond. Math. Soc. (2) 62, 27-40 (2000; Zbl 0964.11016).]
Let \(A, B\) be subsets of a finite commutative group \(G\), and let \( { \mathcal R } \subset A\times B \). The \( { \mathcal R } \)-restricted sum \( A \mathop{+}\limits^{ \mathcal R } B\) is defined as the set of all sums \(a+b\) with \(a\in A\), \(b\in B\), \( (a,b)\not\in \mathcal R \). In the case \( { \mathcal R } = \{ (a,a) \}\) this reduces to the much investigated case of sums with distinct summands.
The author finds estimates for \( |A \mathop{+}\limits^{ { \mathcal R } } B |\) when \( |A |\), \( |B |\) and \( |{ \mathcal R } |\) are given. In case of a cyclic group of prime order he finds essentially the best possible estimates. These results do not yield the Dias da Silva-Hamidoune theorem on distinct summands, where the bound is \( |A |+ |B |-3\), and it is shown by examples that such a strong estimate does not hold in general even when \( { \mathcal R } \) is assumed to be of the special form \( { \mathcal R } = \{(a, \tau (a) \}\) with some injective function \(\tau \).

Reviewer: I.Z.Ruzsa (Budapest)

MSC:

11B75	Other combinatorial number theory
05D99	Extremal combinatorics
20F99	Special aspects of infinite or finite groups

Keywords:

sumsets; restricted set addition; cyclic group of prime order

Citations:

Zbl 0964.11016