Siegel measures
type: Item
instance of: scholarly article
title: Siegel measures
zbMATH Open document ID: 0922.22003
DOI: 10.2307/121033
author: William A. Veech
published in: Annals of Mathematics. Second Series
publication date: 1999-10-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/9811183
full work available at URL: https://eudml.org/doc/119951
review text: The results of the present paper represent a middle ground of sorts between the general Chebyshev theorem of Mazur and the restricted prime geodesic theorem of Veech. Let \(G_N=SL(N,\mathbb{R})\) and \(T_n=SL(N,\mathbb{Z})\). Equip \(G_N/T_N\) with its normalised Haar measure \(\mu_N\). If \(\psi\geq 0\) is a Borel function on \(R^N\) define \(\widehat\psi\) by \(\widehat\psi(gT_N)=\sum_{\vartheta\in\mathbb{Z}^N\setminus\{0\}}\psi(g\cdot\vartheta)\). Define \(M_N\) to be the set of Borel measures \(\upsilon\) on \(R^N\) such that \(M(\upsilon)<\infty\) and \(N_\nu(R)=\upsilon(B(0,R))\). The main theorem for Siegel measures is: If \(\mu\) is a Siegel measure, then there exists a constant \(c(\mu)<\infty\) such that: (i) If \(\psi\geq 0\) is a Borel function, then \(\int_{M_N}\widehat\psi(\upsilon)\mu(d\upsilon)=c(\mu)\int_{\mathbb{R}_N}\psi(u)du\). (ii) If \(\sigma_N\) is the area of the unit sphere in \(\mathbb{R}^N\), then \(\lim_{R\to\infty}\|{N_\nu(R)\over R^N}-{c(\mu)\sigma_N\over N}\|_1=0\). (iii) If \(\mu\) is supported on \(M_N^e=\{\upsilon\in M_N\mid\upsilon(-U)=\upsilon(U)\), \(U\) Borel\} then for all \(\psi\in C_c(R^N)\) \[ \lim_{R\to\infty}\left\|{1\over R^N}\int_{\mathbb{R}^N}\psi\left({\upsilon\over R}\right)\nu(d\upsilon)-C(\mu)\int_{\mathbb{R}^N}\psi(u)du\right\|_1=0. \] If \(N>2\), and if \(\mu\) is such that \(\widehat\psi\in L^2(\mu)\) for all \(\psi\in C_c(\mathbb{R}^N)\) then convergence in (ii) and (iii) also holds pointwise a.e. \(\mu\). As a consequence the main result for quadratic differentials is obtained. The restricted prime geodesic theorem of Veech along with a consequence is proved using the equidistribution theorem of \textit{A. Eskin} and \textit{C. McMullen} [Duke Math. J. 181-209 (1993; Zbl 0798.11025)]. It is also proved that if \(\mu\) is a Siegel measure, there exists \(C\geq 0\) and a singular Siegel measure \(\mu^s\) such that \(\mu=S^C*\mu^s\), where \(S^C:M_N\to M_N\) is defined by \(S^C\upsilon=\upsilon+cm\) and \(m=m(du)\…
reviewed by: Ranjeet Sehmi
Mathematics Subject Classification ID: 22D40
Mathematics Subject Classification ID: 11H99
zbMATH DE Number: 1267821
zbMATH Keywords: Chebyshev theorem
zbMATH Keywords: restricted prime geodesic theorem
zbMATH Keywords: normalised Haar measure
zbMATH Keywords: Borel function
zbMATH Keywords: Siegel measures
zbMATH Keywords: Siegel measure
MaRDI profile type: Publication
OpenAlex ID: W3037513664
arXiv ID: math/9811183
description: scientific article