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Zbl 1162.11318
Lev, Vsevolod F.
Restricted set addition in abelian groups: results and conjectures. (English)
[J] J. Théor. Nombres Bordx. 17, No. 1, 181-193 (2005). ISSN 1246-7405

Summary: We present a system of interrelated conjectures which can be considered as restricted addition counterparts of classical theorems due to Kneser, Kemperman, and Scherk. Connections with the theorem of Cauchy-Davenport, conjecture of Erdös-Heilbronn, and polynomial method of Alon-Nathanson-Ruzsa are discussed. The paper assumes no expertise from the reader and can serve as an introduction to the subject. The final unifying conjecture reads: Conjecture 5.1: For any finite non-empty subsets $A$ and $B$ of an abelian group we have $$\vert A\dot+B\vert\geq\min\{\vert A+B\vert, \vert A\vert+\vert B\vert-1\}-2.$$ In other words, if $\vert (A+B)\setminus(A\dot+B)\vert\geq 3$, then $\vert A\dot+B\vert\geq\vert A\vert+\vert B\vert-3$.

MSC 2000:: *11B75 Combinatorial number theory
20K01 Finite abelian groups
11P70 Inverse problems of additive number theory

Keywords: Restricted set addition; Kneser theorem; Kemperman-Scherk theorem

Cited in: Zbl 1173.11312

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