Hyper-Atoms Applied to the Critical Pair Theory


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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Correspondence to Yahya O. Hamidoune.

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Yahya O. Hamidoune passed away on March 11, 2011 before the final revisions for the already submitted manuscript could be completed. Final revisions have been made post mortem by David J. Gynkiewicz, Department of Mathematical Sciences, University of Memphis, Memphis TN, 38152, USA

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Hamidoune, Y.O. Hyper-Atoms Applied to the Critical Pair Theory. Combinatorica 39, 1281–1315 (2019). https://doi.org/10.1007/s00493-019-2429-5

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