Diophantine approximation and self-conformal measures
type: Item
instance of: scholarly article
title: Diophantine approximation and self-conformal measures
zbMATH Open document ID: 1071.11047
author: Mariusz Urbański
published in: Journal of Number Theory
publication date: 2005-04-07T00:00:00Z
review text: The author considers metric Diophantine approximation using some novel measures on sets of a fractal character. A point \({\mathbf x}= (x_1,\dots, x_n)\in\mathbb{R}^n\) is said to be very well approximable (VWA) if for some \(\delta> 0\), there are infinitely many \({\mathbf p}= (p_1,\dots, p_n)\in\mathbb{Z}^n\) and \(q\in\mathbb{N}\) such that \[ |q{\mathbf x}-{\mathbf p}|_\infty= \max\{|qx_i- p_i|: i= 1,\dots, n\}\leq q^{-{1\over n}-\delta} \] (the notation \(\| q{\mathbf x}-{\mathbf p}\|\) used by the author presumably means the \(L^2\) norm but for the sets involved, it is immaterial whether this or the more usual \(L^\infty\) norm is used). The point \({\mathbf x}\) is multiplicatively VWA if the weaker product inequality \[ \prod^n_{i=1} |qx_i- p_i|\leq q^{-1-\delta} \] holds infinitely often. Khintchine's theorem on metric Diophantine approximation in \(\mathbb{R}^n\) implies that almost all points are not VWA (i.e., the set of VWA points is a set of Lebesgue measure zero) or that \(\mathbb{R}^n\) is extremal and \textit{P. X. Gallagher} [J. Lond. Math. Soc. 37, 387--390 (1962; Zbl 0124.02902)] proved that almost no points are multiplicatively VWA or that \(\mathbb{R}^n\) is strongly extremal. The Hausdorff dimension of the set of VWA points is \(n(n+ 1)/(n+ 1 +\delta)\) [\textit{V. Jarnik}, Math. Z. 33, 505--543(1931; Zbl 0001.32403)] and not \(n\) as stated; that of the multiplicative VWA set is \(n- 1+ 2/(2+ \delta)\) [\textit{J., D. Bovey} and \textit{M. M. Dodson}, Bull. Lond. Math. Soc. 10, 213--218 (1978; Zbl 0384.10013)]. \textit{D. Y. Kleinbock}, \textit{E. Lindenstrauss} and \textit{B. Weiss} studied an extension of this notion to measures [On fractal measures and Diophantine approximation, Sel. Math. (N.S.), 10, No. 4, 479--523 (2004; Zbl 1130.11039)] a Borel measure \(\mu\) on \(\mathbb{R}^n\) is said to be (strongly) extremal if \(\mu\)-almost all points are not (multiplicatively) VWA; friendly measures are introduced and shown to be strongly ext…
Mathematics Subject Classification ID: 11J83
Mathematics Subject Classification ID: 11K60
Mathematics Subject Classification ID: 37B10
zbMATH DE Number: 2153780
zbMATH Keywords: Diophantine approximation
zbMATH Keywords: Conformal measure
zbMATH Keywords: Gibbs state
zbMATH Keywords: Extremal measure
zbMATH Keywords: Absolutely friendly measure
zbMATH Keywords: Conformal iterated function system
zbMATH Keywords: Hausdorff measure
zbMATH Keywords: Hölder families of functions
reviewed by: M. M. Dodson
MaRDI profile type: Publication
full work available at URL: j.jnt.2004.07.004
OpenAlex ID: W2073582262
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DOI: 10.1016/J.JNT.2004.07.004
description: scientific article; zbMATH DE number 2153780