On the discrepancy of (n\(\alpha\) ). II
type: Item
instance of: scholarly article
title: On the discrepancy of (n\(\alpha\) ). II
zbMATH Open document ID: 0588.10058
DOI: 10.1016/0022-314X(86)90057-0
author: Johannes Schoissengeier
published in: Journal of Number Theory
publication date: 1986-01-01T00:00:00Z
review text: The author gives strong results on the asymptotic behaviour of the discrepancy of the sequence (n\(\alpha)\) for a certain class of \(\alpha\) 's. Using a fundamental result of his recent paper [Acta Arith. 44, 241- 279 (1984; Zbl 0506.10031)] he first gives (possibly) best lower estimates for arbitrary irrational \(\alpha\) (with an absolute constant not depending on \(\alpha)\) and then also gives sharp upper estimates for \(\alpha =[a_ 0;a_ 1,a_ 2,...],\) such that \(a_ i\) is even for \(i\geq 1.\) If in addition the sequence \((a_ k)\) is Cesarò-bounded then a very explicit formula for the asymptotic behaviour of the discrepancy is obtained. This is a very significant generalization of the earlier interesting special cases \(\alpha =(1+\sqrt{5})/2\) (Dupain) and \(\alpha =\sqrt{2}\) (Dupain, Sós) [see the paper of \textit{L. Ramshaw}, J. Number Theory 13, 138-175 (1981; Zbl 0458.10035)].
Mathematics Subject Classification ID: 11K06
Mathematics Subject Classification ID: 11J71
zbMATH DE Number: 3943954
zbMATH Keywords: asymptotic behaviour
zbMATH Keywords: discrepancy
zbMATH Keywords: lower estimates
zbMATH Keywords: upper estimates
zbMATH Keywords: Cesarò-bounded
MaRDI profile type: Publication
cites work: Extension à la dimension n d'un théorème de Ortel et Schneider. (Extension of a theorem of Ortel and Schneider to n dimensions)
cites work: On the discrepancy of the sequence formed by the multiples of an irrational number
cites work: Q3337566
cites work: Q4039979
full work available at URL: 0022-314x(86)90057-0
OpenAlex ID: W2071200011
description: scientific article; zbMATH DE number 3943954