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Subsets with small sums in abelian groups. I: The Vosper property. (English) Zbl 0883.05065
This paper characterizes the finite subsets $$B$$ in an abelian group $$G$$ such that, for any finite subset $$A$$ having at least two elements, $$|A + B|\geq$$ min$$(|G|-1, |A|+|B|)$$. The approach uses graph-theoretic ideas on Cayley graphs, including the concepts of “fragments” and “atoms” in a graph (as in H. A. Jung [Math. Ann. 202, 307-320 (1973; Zbl 0239.05133)]). Applications are given to diagonal forms over finite fields and to characterizing Cayley graphs (or loop networks) of high connectivity.

##### MSC:
 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 05C40 Connectivity 20D60 Arithmetic and combinatorial problems involving abstract finite groups
##### Keywords:
abelian group; Cayley graph; connectivity; fragment; atom
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