Ergodic theory on \(SL(n)\), diophantine approximations and anomalies in the lattice point problem
The appearance of anomalously small, particularly logarithmically small, errors in the lattice point problem for polyhedra in \(\mathbb{R}^n\) is studied in the paper. The problem is reduced to simultaneous diophantine approximation for systems of linear forms associated with the set of faces of a given polyhedron \(P\subset \mathbb{R}^n\). Next these diophantine approximations are interpreted in terms of certain flows of diagonal matrices on the homogeneous space of the \(n\)-dimensional unimodular lattices \({\mathfrak L}_n =SL(n, \mathbb{R})/SL (n,\mathbb{Z})\), and methods of ergodic theory are applied to the study of the behavior of the error for the polyhedron. The main results given in the paper can be stated as follows. Let \(tP\) be the dilatation of a compact polyhedron \(P\subset \mathbb{R}^n\) by a factor \(t>0\), \(N(tP, \Gamma)\) denote the number of points of a unimodular lattice \(\Gamma \subset {\mathfrak L}_n\) falling into \(tP\), and \(\varepsilon >0\) be arbitrarily small. Then for almost every \(\Gamma \in {\mathfrak L}_n\) (with respect to the unique invariant probability measure on \({\mathfrak L}_n)\) one has the following asymptotics as \(t\to \infty\) \[ N(tP, \Gamma)= t^n \text{vol} P+ O\bigl( (\log t)^{n-1+ \varepsilon} \bigr). \] For almost every rotation \(U\in SO(n)\) (with respect to the Haar measure on \(SO(n))\), the error is also dominated by a power of \(\log t\), \[ N(tUP, \Gamma) = t^n \text{vol} P +O\bigl( (\log t)^{2(n-1) +\varepsilon} \bigr). \] Finally, let \(P_k \subset \mathbb{R}^n\) be a compact polyhedron such that all of its vertices have coordinates belonging to a real algebraic number field \(k\), then \[ N(tP_k, \Omega_k) =t^n \text{vol} P_k +O (t^\varepsilon) \] for certain algebraic lattices \(\Omega_k \in {\mathfrak L}_n\) explicitly constructed in terms of the field \(k\). Moreover, the paper also contains a series of new metric results in the geometry of numbers. Previously a part of the results of the present paper has been stated in [Dokl. Ross. Akad. Nauk 355, No. 5, 609-611 (1997)]. \{In the references and in the text E. Krätzel is stated without umlaut and his entries are not in the right order\}.
scientific article; zbMATH DE number 1186155