On fractal measures and Diophantine approximation
type: Item
instance of: scholarly article
title: On fractal measures and Diophantine approximation
zbMATH Open document ID: 1130.11039
author: Dmitry Kleinbock
author: Elon Lindenstrauss
author: Barak Weiss
published in: Selecta Mathematica. New Series
publication date: 2005-05-13T00:00:00Z
review text: The paper under review is concerned with an extension of Mahler's problem in metric number theory, originally resolved by \textit{V. G. Sprindzuk} [''Mahler's problem in metric number theory,'' Nauka i Teknika, Minsk (1967; Zbl 0168.29504)]. Mahler conjectured that almost no points with respect to the natural measure on the curve \((x, x^2, \dots, x^n)\) should be no better approximable by rationals with the same denominator than is the case for generic points in \(\mathbb R^n\). Here the approximation is measured in the sup-norm. A subset \(E\) of \(\mathbb R^n\) supporting a natural measure is said to be \textit{extremal} if it satisfies this property. If we instead consider multiplicative approximation, where the distances of the individual coordinates are multiplied, we arrive at the notion of \textit{strong extremality}, which implies extremality. In a far-reaching extension of Mahler's conjecture, \textit{D. Y. Kleinbock} and \textit{G. A. Margulis} [Ann. Math. (2) 148, No.1, 339--360 (1998; Zbl 0922.11061)] showed that manifolds satisfying a certain non-degeneracy condition are strongly extremal. The present paper extends and resolves Mahler's conjecture further. We say that a measure is (strongly) extremal if the support of the measure satisfies the (strong) extremality property. A measure \(\mu\) on \(\mathbb R^n\) is said to be \textit{friendly} if it is doubling (or Federer) almost everywhere, if the \(\mu\)-measure of any affine hyperplane is zero, and if the measure satisfies a certain technical decay condition. The main result of the paper under review is that friendly measures are strongly extremal. The class of friendly measures include the volume measure on non-degenerate manifolds, and so the result of \textit{Kleinbock} and \textit{Margulis} [loc.cit.] is contained in the main result. In fact, this is part of a larger class of push-forwards of the so-called absolutely friendly measures. Other examples are also given. The Hausdorff \(s\)-measure re…
reviewed by: Simon Kristensen
Mathematics Subject Classification ID: 11J83
zbMATH DE Number: 2168146
zbMATH Keywords: Diophantine approximation
zbMATH Keywords: extremality
zbMATH Keywords: friendly measures
zbMATH Keywords: dynamical systems.
MaRDI profile type: Publication
full work available at URL: s00029-004-0378-2
OpenAlex ID: W4246396013
DOI: 10.1007/S00029-004-0378-2
description: scientific article; zbMATH DE number 2168146