Restricted rational approximation and Apéry-type constructions
type: Item
instance of: scholarly article
title: Restricted rational approximation and Apéry-type constructions
zbMATH Open document ID: 1198.11057
DOI: 10.1016/S0019-3577(09)00020-2
published in: Indagationes Mathematicae. New Series
publication date: 2010-04-08T00:00:00Z
review text: Some definitions: Let us denote by \(\mathcal E\) the set of all functions \(\phi:\mathbb N^*\rightarrow\mathbb N^*\) with \(\mathbb N^*=\{1,2,3,\dots\}\) such that: 1) For any \(q\geq 1\), \(\phi(q+1)\) is a multiple of \(\phi(q)\). 2) The limit \(\gamma_\phi:=\lim_{q\rightarrow\infty}\frac{\log\;\phi(q)}{\log\;q}\) exists and satisfies \(0\leq \gamma_\phi<1\). For \(\phi\in\mathcal E\) and \(\xi\in\mathbb R\backslash \mathbb Q\), the \textit{\(\phi\)-exponent of irrationality} of \(\xi\) is the supremum denoted by \(\mu_\phi(\xi)\), of the set of real numbers \(\mu\) for which there are infinitely many \(q\geq 1\) such that \(q\) is a multiple of \(\phi(q)\) and \(|\xi-\frac{p}{q}|\leq \frac{1}{q^\mu}\) for some \(p\in\mathbb Z\). Observe that this definition generalizes the usual exponent of irrationality \(\mu(\xi)\). In this article the author obtains some general results on this exponent(a lower bound, the Hausdorff dimension of the set where it is large, connections to some linear linear form in \(1\) and \(\xi\), \(\dots\)). \textit{The context:} \textit{R. Apéry} has proved [Astérisque 61, 11--13 (1979; Zbl 0401.10049)] that, for \(\xi=\zeta(3), \alpha=e^3(1+\sqrt{2})^{-4}<1\), and \(\beta=e^3(1+\sqrt{2})^4>1\), the following holds: S.1: There exist two integer sequences \((u_n)_{n\geq 1}\) and \((v_n)_{n\geq 1}\) such that \(u_n\geq 0, |u_n\xi-v_n|^{1/n}\rightarrow \alpha\) and \(u_n^{1/n}\rightarrow \beta\). S.2: For \(d_n:=lcm(1,2,\dots,n)\), \(\delta_n:=d_n^3\) divides \(u_n\) for any \(n\geq 1\). S.3: S.1 implies \(\mu(\xi)\leq 1- \frac{\log\;\beta}{\log\;\alpha}\), where \(\mu(\xi)\) is the coefficient of irrationality of \(\xi\), so that \(\mu(\zeta(3))\leq 13.4178202\cdots\). S.4: \textit{S. Fischer, T. Rivoal} [Proc. Am. Math. Soc. 138, No. 3, 799--808 (2010; Zbl 1222.11094)] have proved that, for \(\xi\in \mathbb R\backslash\mathbb Q\), if \(\mu(\xi)\leq 1-\frac{\log\;\beta}{\log\;\alpha}\) with \(0<\alpha<1<\beta\), then \(S.1\) holds. \textit{Th…
reviewed by: Roland Quême
Mathematics Subject Classification ID: 11J17
Mathematics Subject Classification ID: 11J82
zbMATH DE Number: 5691187
zbMATH Keywords: exponent of irrationality
zbMATH Keywords: restricted rational approximation
zbMATH Keywords: Hausdorff dimension
author: Stéphane Fischler
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5691187