On irregularities of distribution of weighted sums-of-digits
type: Item
instance of: scholarly article
title: On irregularities of distribution of weighted sums-of-digits
zbMATH Open document ID: 1217.11010
author: Gerhard Larcher
author: H. Zellinger
published in: Discrete Mathematics
publication date: 2011-01-07T00:00:00Z
review text: Let \(s(n)\) be the sum-of-digits function of \(n\) in base \(2\). \textit{A. O. Gelfond} [Acta Arith. 13, 259--265 (1968; Zbl 0155.09003)] has shown that \((s(3n))_{n\geq 0}\) is uniformly distributed in the residue classes modulo \(2\). However, \textit{D. J. Newman} [Proc. Am. Math. Soc. 21, 719--721 (1969; Zbl 0194.35004)] was able to prove that \[ \#\{0\leq n<N: s(3n)\equiv 0\bmod 2\}-\frac N2>cN^{\frac{\log 3}{\log 4}}\quad\text{ for all }N. \] This motivated the authors of the paper under review to consider whether also the weighted sum-of-digits function satisfies a Newman-type property. To this end let \(\pmb{\gamma}=(\gamma_0,\gamma_1,\ldots)\in\{0,1\}^{\mathbb{N}_0}\) be a weight sequence. Then the weighted sum-of-digits function \(s_\gamma(n)\) is given by \[ s_\gamma(n)=a_r\gamma_r+\cdots+a_1\gamma_1+a_0\gamma_0 \quad\text{for}\quad n=a_r2^r+\cdots+a_12+a_0. \] Furthermore for a given weight sequence \(\pmb{\gamma}\), and a residue class \(k\in\{0,1,2\}\), we say that \((s_\gamma(3n+k))_{n\geq0}\) satisfies a strong Newman-type property if for some \(c,\kappa>0\) we have \[ \#\{0\leq n<N:s(3n+k)\equiv 0\bmod 2\}-\frac N2>cN^{\kappa} \] or \[ \#\{0\leq n<N:s(3n+k)\equiv 0\bmod 2\}-\frac N2<-cN^{\kappa} \] for all \(N\in\mathbb{N}_0\). Similarly, we say that the sequences \((s_\gamma(3n+k))_{n\geq0}\) satisfies a weak Newman-type property if the above inequalities only hold for sufficiently large \(N\). The authors of the present paper give a complete characterization of weight sequences together with residue classes which satisfy strong, weak or no Newman-type properties. These results have applications in the investigation of the distribution of Niederreiter--Halton sequences.
reviewed by: Manfred G. Madritsch
Mathematics Subject Classification ID: 11A63
Mathematics Subject Classification ID: 11K38
zbMATH DE Number: 5833544
zbMATH Keywords: sum-of-digits
zbMATH Keywords: weighted sum-of-digits
zbMATH Keywords: irregularities of distribution
zbMATH Keywords: Newman's phenomenon
zbMATH Keywords: distribution of Niederreiter--Halton sequences
MaRDI profile type: Publication
full work available at URL: j.disc.2010.09.020
OpenAlex ID: W1971814352
cites work: A summation formula related to the binary digits
cites work: Newman's phenomenon for generalized Thue-Morse sequences
cites work: Sur les nombres qui ont des propriétés additives et multiplicatives données
cites work: Q3505067
cites work: On the distribution properties of Niederreiter-Halton sequences
cites work: DISTRIBUTION PROPERTIES OF GENERALIZED VAN DER CORPUT–HALTON SEQUENCES AND THEIR SUBSEQUENCES
cites work: On existence and discrepancy of certain digital Niederreiter–Halton sequences
cites work: Average growth-behavior and distribution properties of generalized weighted digit-block-\-counting functions
cites work: On the Number of Binary Digits in a Multiple of Three
cites work: Q4003879
cites work: Q5449221
DOI: 10.1016/J.DISC.2010.09.020
Recommended article: On a conjecture of Dekking: the sum of digits of even numbers
Recommended article: Q3222913
Recommended article: On a problem of Gelfond: the sum of digits of prime numbers
Recommended article: The sum of digits of \lfloor nc\rfloor
Recommended article: The distribution of generalized sum-of-digits functions in residue classes
Recommended article: Generalized Newman phenomena and digit conjectures on primes
Recommended article: Q3505067
Recommended article: Integers with an average sum of digits
Recommended article: On the distribution of the truncated sum-of-digits function of polynomial sequences in residue classes
Recommended article: A remark on a summation formula related with the sum of digits
description: scientific article; zbMATH DE number 5833544