Growth of products of subsets in finite simple groups (Q6593642)

Let \(G\) be a finite group and let \(A,B \subseteq G\). The problem of studying the growth of the product \(AB=\{ab \mid a \in A, B \in B \}\) in \(G\) is to determine limitations to \(|AB|\) as a function of \(|A|\) and \(|B|\). Part of the interest of this problem revolves around a conjecture of \textit{M. W. Liebeck} et al. [Bull. Lond. Math. Soc. 44, No. 3, 469--472 (2012; Zbl 1250.20018)], which claims that for any finite simple non-abelian group \(G\) and any \(A \subseteq G\) with \(|A|\geq 2\), \(G\) can be written as the product of \(N\) conjugates of \(A\) with \(N=O(\log |G|/ \log |A|)\). This conjecture generalizes a result of \textit{M. W. Liebeck} and \textit{A. Shalev} [Ann. Math. (2) 154, No. 2, 383--406 (2001; Zbl 1003.20014)], which proves it for \(A\) a normal subset, that is, a union of conjugacy classes of \(G\).\N\NIn attempting to prove the conjecture, or partial cases thereof, a natural way is to show that the product of two subsets has size comparable to the product of the sizes of the two original sets. A result in this vein is the main theorem of the paper under review (Theorem 1.1): For any \(\varepsilon > 0\), there exists \(\delta > 0\) such that if \(G\) is a finite simple non-abelian group, \(A\) is a subset and \(B\) is a normal subset with \(|A|, |B| \leq |G|^{\delta}\), then \(|AB| \geq |A||B|^{1-\varepsilon}\).\N\NTheorem 1.1 improves the main result of [\textit{M. W. Liebeck} et al., Trans. Am. Math. Soc. 369, No. 12, 8765--8779 (2017; Zbl 1476.20018)].