On the asymptotic distribution of integer matrices
type: Item
instance of: scholarly article
title: On the asymptotic distribution of integer matrices
zbMATH Open document ID: 1284.11100
author: Andrey Illarionov
published in: Moscow Journal of Combinatorics and Number Theory
publication date: 2012-09-05T00:00:00Z
reviewed by: Roland Quême
Mathematics Subject Classification ID: 11J71
Mathematics Subject Classification ID: 11J70
zbMATH DE Number: 6077916
zbMATH Keywords: distribution of integer matrices
zbMATH Keywords: local minimum
zbMATH Keywords: multidimensional continued fraction
zbMATH Keywords: average length of continued fraction
MaRDI profile type: Publication
Recommended article: Asymptotic distribution of \(n\)th-order integral matrices belonging to a given residue class
Recommended article: Q3947739
Recommended article: Q3963099
Recommended article: Q4206166
Recommended article: Asymptotics of the distribution of the spectrum of random matrices
Recommended article: Asymptotics for incidence matrix classes
Recommended article: Asymptotic properties of random matrices and pseudomatrices
Recommended article: Q5558018
Recommended article: Asymptotics of random matrices and matrix valued processes
Recommended article: The asymptotic distribution of the condition number for random circulant matrices
review text: Let \(s, N\geq 2\) be integers, \(M_s(\mathbb Z), M_s(\mathbb R)\) square matrices of order \(s\), \(M_s(\mathbb Z,N):=\{X\in M_s(\mathbb Z)\;|\;\det X=N\}\), \(TM_s(\mathbb Z):=\{((t_{ij}))\in M_s(\mathbb Z)\;|\;t_{ij}=0\) for \(j>i,\;0\leq t_{ij} < t_{ii}\) for \(j\leq i,\;i=1,\dots,s\}\), \(TM_s(\mathbb Z,N):=TM_s(\mathbb Z)\cap M_s(\mathbb Z,N)\), \(\mathrm{GL}_s(\mathbb R):=\{X\in M_s(\mathbb R) \;|\;\det X\not=0\}\). Let the set \(\Omega\subset \mathrm{GL}_s(\mathbb R)\) satisfying the two conditions:NEWLINENEWLINE(A) \(\Omega=\{((x_{ij}))\;|\;(x_{i1},\dots,x_{is})\in V_i,\;i=1,\dots,s\}\), where \(V_i, i=1,\dots,s\) is a connected cone in \(\mathbb R^s\) with the vertex at the point \(x=0\), the boundary of \(V_i\) is piecewise differentiable (an hypersurface \(S\subset \mathbb R^s\) is \textit{piecewise differentiable} if \(S\) consists of a finite number of differentiable hypersurfaces).NEWLINENEWLINE(B) There exists a positive constant \(C\) such that \(\prod_{i=1}^s\max_{1\leq j\leq s}|x_{ij}|\leq C\cdot\det X\) for any \(X=((x_{ij}))\in\Omega\).NEWLINENEWLINEThe main aim of the article is to obtain an asymptotic formula for the number of integer matrices \(M\in \Omega\) with \(\det M=N\). NEWLINELet \(\mathcal D_s(\mathbb R_+)\) be the set of diagonal matrices \(((x_{ij}))\in \mathrm{GL}_s(\mathbb R)\) such that \(x_{ii}\in \mathbb R^+,\;i=1,\dots,s\). Consider the \(s(s-1)\)-dimension manifold \(\mathrm{PGL}_s(\mathbb R)=\mathcal D_s(\mathbb R^+)\backslash \mathrm{GL}_s(\mathbb R)\) (the set of right cosets). Let \(\mathcal P(\Omega)\) be the image of \(\Omega\in \mathrm{GL}_s(\mathbb R)\) under the projection \(\mathrm{GL}_s(\mathbb R)\rightarrow \mathrm{PGL}_s(\mathbb R)\).NEWLINENEWLINEThe author defines a certain measure \(\mu\) on \(\mathrm{PGL}_s(\mathbb R)\): suppose that \(k=\{k_1,\dots,k_s\}\) is a permutation of \((1,\dots,s)\) and \(\theta=(\theta_1,\dots,\theta_s), \;\theta_i=\pm 1\). Let \(\mathrm{GL}_s(\mathbb R,k,\theta):=\{((x_{ij}))\in …
description: scientific article; zbMATH DE number 6077916