On sums of fractional parts \(\{na+y\}\)
type: Item
instance of: scholarly article
title: On sums of fractional parts \(\{na+y\}\)
zbMATH Open document ID: 0886.11045
published in: Journal of Number Theory
publication date: 1997-07-24T00:00:00Z
review text: The authors obtain an explicit formula for the sum \[ C_m(\alpha,\gamma)= \sum_{1\leq k\leq m} (\{k\alpha+ \gamma\}-\textstyle{{1\over 2}}), \] where \(\alpha\) is an irrational. The formula is based on the continued fraction expansion of \(\alpha\), and the method is very related to \textit{J. Schoißengeier} [Acta Arith. 44, 241-279 (1984; Zbl 0544.10035); see also Monatsh. Math. 102, 59-77 (1986; Zbl 0613.10011); and Math. Ann. 296, 529-545 (1993; Zbl 0786.11043)] and \textit{C. Baxa} and \textit{J. Schoißengeier} [Acta Arith. 68, 281-290 (1994; Zbl 0828.11038)]. In the present paper, tight bounds for \(C_m(\alpha,\gamma)\) are established in terms of the generalized Zechendorf-expansion (= Ostrowski-expansion) related to \(\alpha\). Furthermore, an explicit formula is stated which immediately yields bounds for the discrepancy of the \((\{n\alpha\})^\infty_{n= 1}\) sequence.
reviewed by: Robert F. Tichy
Mathematics Subject Classification ID: 11K06
Mathematics Subject Classification ID: 11J70
Mathematics Subject Classification ID: 11K38
Mathematics Subject Classification ID: 11J71
zbMATH DE Number: 1037291
zbMATH Keywords: continued fraction expansion
zbMATH Keywords: generalized Zechendorf-expansion
zbMATH Keywords: Ostrowski-expansion
zbMATH Keywords: discrepancy
author: Chris Pinner
MaRDI profile type: Publication
full work available at URL: jnth.1997.2080
OpenAlex ID: W1967818498
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DOI: 10.1006/JNTH.1997.2080
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description: scientific article; zbMATH DE number 1037291