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On a nonabelian Balog-Szemerédi-type lemma. (English) Zbl 1223.11014
Let \(G\) be a group and \(A\subset G\) a finite subset such that \(|A^2|\leq K|A|\) Thus if \(A\) is a collection of free generators then \(K=|A|\). On the other side, if \(K=1\) then \(A\) is a coset of a subgroup of \(A\). The question is the extent to which a structure persists under the condition that \(K\) is slightly larger than \(1\), say \(O(1)\) as \(|A|\to\infty\). The author proves (thereby removing the \(\varepsilon\)-dependence in T. Tao’s Proposition C3 of [Freiman’s theorem for solvable groups. arXiv:0906.3535 (2009)] that there is a symmetric neighbourhood of the identity \(S\) such that \(S^k\subset A^2A^{-1}\) and \(|S|\geq \exp(-K^{O(k)})|A|\), where \(k\in {\mathbb Z}\) is a given parameter. Here a set \(S\subset G\) is called a symmetric neighbourhood of the identity if \(1_G\in S\) and \(S=S^{-1}\).

MSC:
11B13 Additive bases, including sumsets
11P99 Additive number theory; partitions
20F99 Special aspects of infinite or finite groups
20P99 Probabilistic methods in group theory
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[1] DOI: 10.1017/CBO9780511755149
[2] Ruzsa, Acta Arith. 60 pp 191– (1991)
[3] DOI: 10.1007/s00493-008-2271-7 · Zbl 1254.11017
[4] DOI: 10.1112/jlms/jdl021 · Zbl 1133.11058
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