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The Structure of Locally Compact Approximate Groups


A subset A of a locally compact group G is said to be a K-approximate group if it contains the identity, is closed under inverses, and if the product set AA = {ab | a,b \in A} can be covered by K left-translates of A. In this work we prove a structure theorem on open precompact K-approximate groups that describes them as a combination of compact subgroups, convex sets in Lie algebras of dimension bounded in terms of K, and certain generalizations of arithmetic progressions. Our results can naturally be regarded as generalizations of a theorem of Breuillard, Green, and Tao that describes finite approximate groups, and also of a theorem of Gleason and Yamabe, which describes locally compact groups as inverse limits of Lie groups.

Along the way we prove continuous analogues of additive combinatorial results of Sanders, Croot, and Sisask, and model-theoretic results of Hrushovski, all of which were first obtained in the finite context.

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