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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

Citations:

Zbl 0948.11008
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