Metrical theorems on restricted Diophantine approximations to points on a curve.
type: Item
instance of: scholarly article
title: Metrical theorems on restricted Diophantine approximations to points on a curve.
zbMATH Open document ID: 1062.11051
author: Glyn Harman
author: Huw Jones
published in: Journal of Number Theory
publication date: 2003-04-02T00:00:00Z
review text: In a previous paper [J. Number Theory 29, 364--375 (1988; Zbl 0655.10049)], the first author proved that for almost all \(\alpha>0\) there are infinitely many primes \(p,q\) such that \(| p\alpha-q| <\psi(p)\), if \(\psi\) satisfies a mild hypothesis and the sum over primes \(\sum\psi(p)(\log p)^{-1}\) diverges. This has been generalized to higher dimensions by the second author [Acta Arith. 99, 205--225 (2001; Zbl 1013.11040)]. In the present paper the authors consider the much more difficult problem of obtaining such approximations to almost all points on a curve in \(\mathbb R^2\). Their main result is: Let \(\varepsilon>0\), \(\tau>1\), and \(\mu=\frac{3}{16}\). Then for almost all \(\alpha>0\) there are infinitely many primes \(p,q,r\) such that \(0<p\alpha-r<p^{-\mu+\varepsilon}\), \(0<p\alpha^\tau-q<p^{-\mu+\varepsilon}.\)
Mathematics Subject Classification ID: 11J83
Mathematics Subject Classification ID: 11K60
Mathematics Subject Classification ID: 11J25
zbMATH DE Number: 1891427
zbMATH Keywords: metric diophantine approximation
zbMATH Keywords: approximation by primes
zbMATH Keywords: approximation on manifolds
zbMATH Keywords: diophantine inequalities
zbMATH Keywords: infinitely many solutions
reviewed by: Veikko Ennola
MaRDI profile type: Publication
full work available at URL: jnth.2002.2781
OpenAlex ID: W1978074074
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DOI: 10.1006/JNTH.2002.2781
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description: scientific article; zbMATH DE number 1891427