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Approximate groups (according to Hrushovski and Breuillard, Green, Tao). (English) Zbl 1358.11024
Séminaire Bourbaki. Volume 2013/2014. Exposés 1074–1088. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-804-6/hbk). Astérisque 367-368, 79-113, Exp. No. 1077 (2015).
Summary: Given a group \(G\), a symmetric subset \(X\) containing the identity is said to be a \(K\)-approximate group if \(XX\) can be covered by at most \(K\) translates \(gX\). Here \(K\) is a positive integer. The main result describes a finite \(K\)-approximate group as being essentially finite-by-nilpotent.
See also the two articles mentioned in the title: E. Hrushovski [J. Am. Math. Soc. 25, No. 1, 189–243 (2012; Zbl 1259.03049)] and E. Breuillard et al. [Publ. Math., Inst. Hautes Étud. Sci. 116, 115–221 (2012; Zbl 1260.20062)].
For the entire collection see [Zbl 1317.00017].

MSC:
11B30 Arithmetic combinatorics; higher degree uniformity
03C98 Applications of model theory
11P70 Inverse problems of additive number theory, including sumsets
03C07 Basic properties of first-order languages and structures
03C50 Models with special properties (saturated, rigid, etc.)
03H05 Nonstandard models in mathematics
20D15 Finite nilpotent groups, \(p\)-groups
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