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

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 $$\tau$$.

MSC:

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

Zbl 0964.11016
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