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

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.

### 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
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