Discretely approximable locally compact groups and Jessen's theorem for nilmanifolds
A topological group \(G\) is said to be \textit{approximable by discrete subgroups}, if there exists a sequence of discrete subgroups \((H_n)_{n\in \mathbb N}\) of \(G\) such that, for any non-empty open set \(O\) of \(G\), there exists an integer \(k\) such that \(O\cap H_n\not= \emptyset\) for every \(n\geq k\). This paper further develops investigation on approximation of Lie groups by discrete subgroups. The main result established by the authors is that the identity component \(G_0\) of a locally compact discretely approximable group \(G\) is nilpotent. In the case of connected nilpotent Lie groups, an explicit approximating sequence of discrete subgroups is given. As an application, an extension is obtained to nilmanifolds of a theorem of \textit{B. Jessen} [Ann. Math. (2) 35, 248--251 (1934; Zbl 0009.30603)] concerning generalized limits of Riemann sums on the 1-dimensional torus \(\mathbb T\).
scientific article; zbMATH DE number 6536787
scientific article