Approximation of \(\lfloor na+s \rfloor\) and the zero of \(\{ na+s \}\)
type: Item
instance of: scholarly article
title: Approximation of \(\lfloor na+s \rfloor\) and the zero of \(\{ na+s \}\)
zbMATH Open document ID: 0823.11037
author: Douglas Bowman
published in: Journal of Number Theory
publication date: 1995-11-06T00:00:00Z
review text: Let \(\alpha\) and \(\beta\) be positive real numbers and \(s\) a real number satisfying \(0\leq s<1\). Let \(\lfloor x\rfloor\) denote the greatest integer \(\leq x\), and \(\{x\}= x- \lfloor x\rfloor\). Define \(\Psi (\alpha, \beta; s)\) to be the least positive integer \(n\) such that \(\lfloor n\alpha+s \rfloor\neq \lfloor n\beta+ s\rfloor\). When \(s=0\), a simple explicit formula for \(\Psi\) is given, and otherwise more complicated formulas are obtained. When \(\alpha\) is irrational, a formula for finding the least \(n\in \mathbb{Z}^ +\) such that \(\{n\alpha+ s\}=0\) is presented. A natural characterization of the approximation properties of intermediate convergents (including convergents) without reference to the apparatus of continued fractions is given. A new characterization of the sequence \(\lfloor n\alpha \rfloor\) for \(n\geq 1\) is found, and \(\lfloor n\alpha+s \rfloor\) for \(n\geq 1\) is also characterized in a different way.
reviewed by: Gerhard Larcher
Mathematics Subject Classification ID: 11J70
zbMATH DE Number: 714114
zbMATH Keywords: generalized congruences
zbMATH Keywords: continued fractions
MaRDI profile type: Publication
full work available at URL: jnth.1995.1008
OpenAlex ID: W2046623466
DOI: 10.1006/JNTH.1995.1008
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description: scientific article; zbMATH DE number 714114