A new critical pair theorem applied to sum-free sets in Abelian groups. (English) Zbl 1045.11072

Hamidoune, Yahya Ould; Plagne, Alain

Comment. Math. Helv. 79, No. 1, 183-207 (2004).

Let \(G\) be a finite Abelian group. A subset \(A\subset G\) is said to be a Vosper subset of \(G\) if for any \(X\subset G\), with \(| X| \geq 2\), we have \(| A+X| \geq \min (| G| -1, | A| +| X| ).\) And if \(H\leq G\), let us define \(\psi_H\) to be the canonical homomorphism from \(G\) onto \(G/H\). In this paper the authors present the following generalization of Vosper’s theorem for Abelian groups.
“Let \(A\) be a generating subset of a finite Abelian group \(G\) such that \(0\in A\) and \(| A| \leq | G| /2\). Then there exists \(H\leq G\) with \(| A+H| \leq \min(| G|, | A| +| H| )\) such that \(\psi_{H}(A)\) is either a Vosper subset or an arithmetic progression in \(G/H\) (i.e., for some pair \(a,d\in G/H\), \(\psi_{H}(A)=\{a+jd \mid j=1,\cdots, s,\) for some natural number \(s\}\)”.
Next the authors apply this theorem to the theory of \((k,l)\)-free sets in finite Abelian groups, describing in many cases the structure and cardinality of the maximal \((k,l)\)-free sets.

Reviewer: Hemar Godinho (Brasilia)

MSC:

11P70	Inverse problems of additive number theory, including sumsets
20D60	Arithmetic and combinatorial problems involving abstract finite groups
20K01	Finite abelian groups
11B25	Arithmetic progressions

Keywords:

additive number theory; Vosper theorem; \((k,l)\)-free sets; finite Abelian groups