# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.