## Restricted set addition in groups. I: The classical setting.(English)Zbl 0964.11016

The paper is devoted to the study of the restricted sumset $$A\hat +A$$ containing the sums $$a+b$$, $$a,b\in A$$, $$a\neq b$$. If only the cardinality of $$A$$ is given, then the optimal estimate is easy for sets of integers, and is known for residues modulo a prime (Dias da Silva and Hamidoune). The author finds an estimate valid for every group, namely $$|A\hat +A |\geq \vartheta |A |-L-2$$, where $$L$$, the “doubling constant” of the group, denotes the maximal number of solutions of the equation $$x+x=a$$, and $$\vartheta =(1+\sqrt 5)/2$$. He conjectures that the optimal value is $$\vartheta =2$$, at least for commutative groups.
For integers, and for residues modulo a prime, better estimates are known which also involve the length of the shortest arithmetic progression containing $$A$$, due to G. A. Freiman, L. Low and J. Pitman [Astérisque 258, 163-172 (1999; Zbl 0948.11008)]. These estimates are also improved in the paper, and in some cases the conjectured optimal value is attained.

### MSC:

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

### Keywords:

restricted sumsets; doubling constant

Zbl 0948.11008
Full Text:
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