Asymptotic approximate groups in virtually nilpotent groups

Abstract: Let \(G\) be a group and let \(A\subseteq G\) be a non-empty subset. For \(r,l\in\mathbb N\), \(A\) is said to be an asymptotic \((r,l)\)-approximate group if there exists \(h_0\in\mathbb N\) such that, for every \(h\ge h_0\), there is a set \(X_h\subseteq G\) with \(|X_h|\le l\) and $A^{rh}\subseteq X_hA^h.$ We study this property for subsets of virtually nilpotent groups and show that every finite non-empty symmetric subset of a virtually nilpotent group is an asymptotic approximate group. More generally, the same conclusion holds for finite sets whose powers contain a symmetric word ball of radius comparable to \(h\). In the setting of infinite sets, we show a restricted nonabelian analogue of the abelian semilinear-set theorem.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)