Hyper-Atoms Applied to the Critical Pair Theory
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The isoperimetric method is often useful for proving results regarding sumsets. Here, we introduce the notion of a hyper-atom into the method, which overcomes a previous weakness when dealing with atoms that are cosets. To show the utility of this new object, we give a new isoperimetric proof of the cornerstone of classical critical pair theory: The Kemperman Structure Theorem, proved in its so-called “dual” formulation.
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The author is grateful to an anonymous referee for many valuable comments on the first two drafts.
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