The distribution of sublattices of \(\mathbb{Z}^m\)
The author counts \(n\)-dimensional sublattices, i.e., \(n\)-dimensional discrete subgroups of \({\mathbb R}^m\). Let \(| \cdot | \) denote the Euclidean norm on \({\mathbb R}^m\). Two lattices \(\Lambda\), \(\Lambda '\) are called similar if there is a linear angle preserving map from \(\Lambda\) to \(\Lambda '\), that is a linear bijection \(\varphi :\Lambda\to \Lambda '\) such that for some \(c>0\) we have \(| \varphi ({\mathbf x})| =c| {\mathbf x}| \) for \({\mathbf x}\in\Lambda\). Let \({\mathcal F}_n\) denote the collection of similarity classes of \(n\)-dimensional sublattices of \({\mathbb R}^m\). As the author points out, \({\mathcal F}_n\) can be interpreted as a fundamental domain for \({\mathcal H}_n/GL_n({\mathbb Z})\) where \({\mathcal H}_n\) is the generalized upper half plane, that is the set of real \(n \times n\)-upper triangular matrices all of whose diagonal elements are positive and of which the element on the upper left corner is equal to \(1\). Such a fundamental domain can be endowed in a natural way with a normalized Borel-measure \(\mu\) with \(\mu ({\mathcal F}_n)=1\). For an \(n\)-dimensional sublattice \(\Lambda\) of \({\mathbb R}^m\) with basis \(\{ {\mathbf x}_1\dots {\mathbf x}_n\} \), we define the determinant of \(\Lambda\) to be the Euclidean length \(| {\mathbf x}_1\wedge\cdots \wedge {\mathbf x}_n| \). For subsets \({\mathcal D}\subset {\mathcal F}_n\) and for \(T>0\), denote by \(N(T,{\mathcal D})\) the number of lattices of which the similarity class belongs to \({\mathcal D}\) and of which the determinant is \(\leq T\). The author shows that for suitable \({\mathcal D}\) one has \(N({\mathcal D},T)\sim c_1(m,n)\mu ({\mathcal D})T^m\) where \(c_1(m,n)\) depends on \(m,n\) only. In a more specific situation, the author gives an asymptotic formula with an error term. \textit{W. Roelcke} [Math. Ann. 131, 260-277 (1956; Zbl 0071.07801)] and \textit{H. Maass} [Math. Ann. 137, 319-327 (1959; Zbl 0085.06902)] had already dealt with the case \(n=2\) using the theory of modular forms. In contrast, the treatment of the author is elementary.
scientific article; zbMATH DE number 1106751