The sum-of-digits-function and uniform distribution modulo 1
type: Item
instance of: scholarly article
title: The sum-of-digits-function and uniform distribution modulo 1
zbMATH Open document ID: 0990.11053
author: Michael Drmota
author: Gerhard Larcher
published in: Journal of Number Theory
publication date: 2002-06-04T00:00:00Z
review text: Let \(s_q(n)\) be the \(q\)-ary sum-of-digits-function for a fixed integer \(q>1\). \textit{M. Mendès France} [J. Number Theory 5, 1-15 (1973; Zbl 0252.10033)] and \textit{J. Coquet} [Acta Arith. 36, 157-162 (1980; Zbl 0429.10032)] showed that the sequence \((\{\alpha\cdot s_q(n)\})\) is uniformly distributed modulo \(1\) for irrational \(\alpha\). \textit{R. F. Tichy} and \textit{G. Turnwald} [J. Number Theory 26, 68-78 (1987; Zbl 0628.10052)] established estimates for the discrepancy of \((\{\alpha\cdot s_q(n)\})\) for irrational \(\alpha\). In the first part of this paper refined estimates for the discrepancy of \((\{\alpha\cdot s_q(n)\})\) is given. In the second part a first result concerning the uniform distribution of \(d\)-dimensional sequences \((\{\alpha_1\cdot s_{q_1}(n)\}, \dots, \{\alpha_d\cdot s_{q_d}(n)\})\) is proved, where \(\alpha_1\), \(\dots\), \(\alpha_d\) are fixed irrational numbers and \(q_1,\dots,q_d>1\) are fixed pairwise coprime integers.
Mathematics Subject Classification ID: 11K06
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 11K38
Mathematics Subject Classification ID: 11K16
zbMATH DE Number: 1749023
zbMATH Keywords: sum-of-digits-function
zbMATH Keywords: uniform distribution modulo \(1\)
zbMATH Keywords: discrepancy
reviewed by: Takao Komatsu
MaRDI profile type: Publication
full work available at URL: jnth.2000.2628
OpenAlex ID: W2008089909
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DOI: 10.1006/JNTH.2000.2628
description: scientific article; zbMATH DE number 1749023