Simultaneous rational approximation via Rickert's integrals
type: Item
instance of: scholarly article
title: Simultaneous rational approximation via Rickert's integrals
zbMATH Open document ID: 1305.41025
author: N. Saradha
author: Divyum Sharma
published in: Indagationes Mathematicae. New Series
publication date: 2014-11-27T00:00:00Z
review text: In the literature simultaneous rational approximation to algebraic numbers has been studied extensively. The authors focus on the method using hypergeometric polynomials as Padé approximants to functions of the form \[ (1+ax)^{\nu},\;a\in\mathbb{Z}\setminus\{0\},\;\nu\in\mathbb Q^{+}\setminus\mathbb N. \] An important tool is a contour integral representation of these approximants due to \textit{J. H. Rickert} [Math. Proc. Camb. Philos. Soc. 113, No. 3, 461--472 (1993; Zbl 0786.11040)]. The authors apply this tool in the following setting. Let \(a\), \(N\) be positive integers, \(0<\nu<1\) rational and write \(\nu=s/n\) with \(\text{gcd}(s,n)=1\). Put \[ e_0=\prod_{p|n}\,p^{\max\{\text{ord}_p(n/2a)+1/(p-1),0\}}, \] \[ (e,e',e'')\;\begin{cases}(2,4/3,168) &\ \text{if}\ n=2\\ (4,3,679) &\ \text{if}\ n=4\\ (e_0,3e_0/2,1) &\ \text{otherwise}\end{cases} \] Theorem 1.1. Let \(\theta_1=(1-{a\over N})^{\nu},\;\theta_2=(1+{a\over N})^{\nu}\) with \(N\geq 6.794 (e')^2a^3\) and assume that at least one of \(\theta_1\), \(\theta_2\) is irrational. Then \[ \max\{\left|\theta_1-{p_1\over q}\right|,\left|\theta_2-{p_2\over q}\right|\}>{c\over 9^{ \lambda}}\eqno{(\ast)} \] for all integers \(p_1\), \(p_2\), \(q\) with \(q>0\), where \[ \lambda=1+{\log{(6\sqrt{3}e'N+12ae')}\over \log{(27(N^2-a^2)/(16a^3e'))}} \] and \[ c={0.005\over ae'e''N}\left(1-{a\over N}\right)^{\nu}\left(1+{2a\over N\sqrt{3}}\right)^{-1-\nu} \left({qNae'' \nu (1-\nu^2)\over 2(N-a)}\right)^{ -\lambda+1}. \] Theorem 1.3. Let \(\theta_1=(1+{a\over N})^{\nu},\;\theta_2=(1+{2a\over N})^{\nu}\) with \(N\geq 8.637 (e')^2a^3\) and assume that at least one of \(\theta_1\), \(\theta_2\) is irrational. Then \((\ast)\) holds with \[ \lambda=1+{\log{(6\sqrt{3}e'N+6ae'(2+\sqrt{3})}\over \log{(27(N-a)(N-2a)/(16a^3e'))}} \] and \[ c={0.005\over ae'e''N}\left(1+{(2+\sqrt{3})a\over N\sqrt{3}}\right)^{-1-\nu} \left({9Nae'' \nu (1-\nu^2)\over 2(N-2a)}\right)^{ -\lambda+1}. \] The third main result (Theorem 1.6) runs along the sam…
reviewed by: Marcel G. de Bruin
Mathematics Subject Classification ID: 41A28
Mathematics Subject Classification ID: 11J68
Mathematics Subject Classification ID: 41A21
Mathematics Subject Classification ID: 33C47
zbMATH DE Number: 6374449
zbMATH Keywords: simultaneous approximation
zbMATH Keywords: Padé approximation
zbMATH Keywords: contour integral
MaRDI profile type: Publication
full work available at URL: j.indag.2014.07.001
OpenAlex ID: W2087880911
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DOI: 10.1016/J.INDAG.2014.07.001
description: scientific article; zbMATH DE number 6374449