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.


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


Zbl 0948.11008
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