Ergodic theory on \(SL(n)\), diophantine approximations and anomalies in the lattice point problem
type: Item
instance of: scholarly article
title: Ergodic theory on \(SL(n)\), diophantine approximations and anomalies in the lattice point problem
zbMATH Open document ID: 0902.11041
published in: Inventiones Mathematicae
publication date: 1998-12-13T00:00:00Z
review text: 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 …
Mathematics Subject Classification ID: 11P21
Mathematics Subject Classification ID: 11J13
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 11H06
Mathematics Subject Classification ID: 22D40
zbMATH DE Number: 1186155
zbMATH Keywords: errors in the lattice point problem
zbMATH Keywords: simultaneous diophantine approximation
zbMATH Keywords: systems of linear forms
zbMATH Keywords: ergodic theory
author: M. M. Skriganov
MaRDI profile type: Publication
full work available at URL: s002220050217
OpenAlex ID: W2002902699
DOI: 10.1007/S002220050217
description: scientific article; zbMATH DE number 1186155