Fourier analysis of continued fractions with restricted quotients
type: Item
instance of: scholarly article
title: Fourier analysis of continued fractions with restricted quotients
zbMATH Open document ID: 1057.11038
author: Martine Queffélec
author: Olivier Ramaré
published in: L'Enseignement Mathématique. 2e Série
publication date: 2004-10-25T00:00:00Z
review text: The main nice result of this paper reads as follows. Let \(A\) be a finite set of positive integers containing at least two elements, and let \(F(A)\) be the set of irrational numbers in \((0,1)\) whose continued fraction expansion has all its partial quotients in \(A\). Then, if the Hausdorff dimension of \(F(A)\) is \(>{1\over 2}\), the set \(F(A)\) contains infinitely many normal numbers. If \(A:= \{1,2,\dots, N\}\) this improves on a result of \textit{R. Kaufman} (more precisely on a remark of Baker to a result of Kaufman) where the result holds true for \(N\geq 3\) (see [Mathematika 27, 262--267 (1980; Zbl 0455.10035)], see also [\textit{H. L. Montgomery}, Ten lectures on the interface between analytic number theory and harmonic analysis. Providence, RI: AMS (1994; Zbl 0814.11001)]). Please note that reference [3] has appeared [\textit{Y. Bugeaud}, C. R. Math., Acad. Sci. Paris 335, No. 2, 117--120 (2002; Zbl 1006.11043)].
Mathematics Subject Classification ID: 11K50
Mathematics Subject Classification ID: 28A78
Mathematics Subject Classification ID: 11K16
Mathematics Subject Classification ID: 42A38
zbMATH DE Number: 2109720
zbMATH Keywords: continued fraction with bounded partial quotients
zbMATH Keywords: normal numbers
zbMATH Keywords: Kaufman measures
reviewed by: Jean-Paul Allouche
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2109720