On semilinear sets and asymptotic approximate groups (Q2122762)

Let \(G\) be a group, \(A \subseteq G\), and \(K \geq 1\). Then \(A\) is a \(K\)-approximate subgroup if there exists \(X \subseteq G\) such that \(A^2 \subseteq XA\). Approximate subgroups have been studied by many authors in the last years. \textit{M. B. Nathanson} [J. Number Theory 191, 175--193 (2018; Zbl 1452.11015)] introduced the notion of asymptotic \(K\)-approximate subgroup, i.e. \(A^n\) is a \(K\)-approximate subgroup for all large \(n\), and proved that if \(G\) is abelian then \(A\) is always an asymptotic \(K\)-approximate subgroup for some \(K\) which depends on \(A\). In the paper under review, the authors give a different proof of this result, and improve the bound on \(K\). They also give a generalization of this result.