Inhomogeneous approximation with coprime integers and lattice orbits
type: Item
instance of: scholarly article
title: Inhomogeneous approximation with coprime integers and lattice orbits
zbMATH Open document ID: 1273.11106
author: Michel Laurent
author: Arnaldo Nogueira
published in: Acta Arithmetica
publication date: 2012-07-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1110.5467
Mathematics Subject Classification ID: 11J20
Mathematics Subject Classification ID: 37A17
Mathematics Subject Classification ID: 11J25
zbMATH DE Number: 6055437
zbMATH Keywords: Diophantine approximation
zbMATH Keywords: primitive points
zbMATH Keywords: density exponents
zbMATH Keywords: ergodic theory
MaRDI profile type: Publication
OpenAlex ID: W2963244361
arXiv ID: 1110.5467
DOI: 10.4064/AA154-4-5
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review text: \textit{J. H. H. Chalk} and \textit{P. Erdős} [Can. Math. Bull. 2, 91--96 (1959; Zbl 0088.25703)] proved: Let \(\xi\) be an irrational real number and let \(y\) be a real number. There exists an absolute constant \(c\) such that the inequality NEWLINE\[NEWLINE|q\xi+p-y|\leq \frac{c(\log\;q)^2}{q(\log \log q)^2},\tag{e1}NEWLINE\]NEWLINE holds for infinitely many pairs of coprime integers \((p,q)\) with \(q>0\). NEWLINENEWLINEThe authors study more generally the Diophantine inequality \(|q\xi+p-y|\leq \psi(|q|)\) for coprime integers \(p\) and \(q\) where \(\psi: \mathbb N\rightarrow \mathbb R^+\) is a given function. At first, they prove:NEWLINENEWLINELet \(\xi\) be an irrational real number and let \(y\) be a nonzero real number. There exist infinitely many integer quadruples \((p_1,q_1,p_2,q_2)\) satisfying \(q_1p_2-p_1q_2=1\) and NEWLINE\[NEWLINE|q_i\xi+p_i-y|\leq \frac{c}{\max (|q_1|,|q_2|)^{1/2}}\leq \frac{c}{\sqrt{|q_i|}}\;(i=1,2)NEWLINE\]NEWLINE with NEWLINE\[NEWLINEc=2\sqrt{3}\max(|1,|\xi|^{1/2}|y|^{1/2}).NEWLINE\]NEWLINE NEWLINEThen, they adress the following problem: Can we replace the function \(\psi(\ell)=\frac{c(\log \ell)^2}{\ell(\log\log \ell)^2}\) occurring in (e1) by a smaller one, possibly \(\psi(\ell)= c\ell^{-1}\)? Towards this problem they prove the following metrical statement.NEWLINENEWLINENEWLINETheorem. Let \(\psi: \mathbb N\to\mathbb R^+\) be a function. Assume that \(\psi\) is non-increasing, tends to \(0\) at infinity and that, for every positive integer \(c\), there exists a positive real number \(c_1\) satisfying \(\psi(c\ell)\geq c_1\psi(\ell), \;\forall \ell\geq1\). Furthermore assume that \(\sum_{\ell\geq 1} \psi(\ell)=+\infty\). Then, for almost all pairs \((\xi,y)\) of real numbers, there exist infinitely many \textit{primitive points} \((p,q)\) (so with \(\gcd(p,q)=1\)) such that NEWLINE\[NEWLINEq\geq 1\text{\;and\;}|q\xi+p-y|\leq \psi(q).\tag{e4}NEWLINE\]NEWLINE If \(\sum_{\ell\geq 1} \psi(\ell)\) converges, the pairs \((\xi, y)\)…
reviewed by: Roland Quême
description: scientific article; zbMATH DE number 6055437