On a conjecture of Dekking : The sum of digits of even numbers

Iurie Boreico[1]; Daniel El-Baz[2]; Thomas Stoll[3]

  • [1] Department of Mathematics Stanford University 450 Serra Mall Stanford, California 94305, USA
  • [2] School of Mathematics University of Bristol University Walk Bristol, BS8 1TW, United Kingdom
  • [3] 1. Université de Lorraine Institut Elie Cartan de Lorraine, UMR 7502 Vandoeuvre-lès-Nancy, F-54506, France 2. CNRS Institut Elie Cartan de Lorraine, UMR 7502 Vandoeuvre-lès-Nancy, F-54506, France

Journal de Théorie des Nombres de Bordeaux (2014)

  • Volume: 26, Issue: 1, page 17-24
  • ISSN: 1246-7405

Abstract

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Let q ≥ 2 and denote by s q the sum-of-digits function in base q . For j = 0 , 1 , ⋯ , q - 1 consider # { 0 ≤ n < N : s q ( 2 n ) ≡ j ( mod q ) } . In 1983, F. M. Dekking conjectured that this quantity is greater than N / q and, respectively, less than N / q for infinitely many N , thereby claiming an absence of a drift (or Newman) phenomenon. In this paper we prove his conjecture.

How to cite

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Boreico, Iurie, El-Baz, Daniel, and Stoll, Thomas. "On a conjecture of Dekking : The sum of digits of even numbers." Journal de Théorie des Nombres de Bordeaux 26.1 (2014): 17-24. <http://eudml.org/doc/275820>.

@article{Boreico2014,
abstract = {Let $q\ge 2$ and denote by $s_q$ the sum-of-digits function in base $q$. For $j=0,1,\dots ,q-1$ consider\[\# \lbrace 0 \le n &lt; N : \;\;s\_q(2n) \equiv j \pmod \{q\} \rbrace .\]In 1983, F. M. Dekking conjectured that this quantity is greater than $N/q$ and, respectively, less than $N/q$ for infinitely many $N$, thereby claiming an absence of a drift (or Newman) phenomenon. In this paper we prove his conjecture.},
affiliation = {Department of Mathematics Stanford University 450 Serra Mall Stanford, California 94305, USA; School of Mathematics University of Bristol University Walk Bristol, BS8 1TW, United Kingdom; 1. Université de Lorraine Institut Elie Cartan de Lorraine, UMR 7502 Vandoeuvre-lès-Nancy, F-54506, France 2. CNRS Institut Elie Cartan de Lorraine, UMR 7502 Vandoeuvre-lès-Nancy, F-54506, France},
author = {Boreico, Iurie, El-Baz, Daniel, Stoll, Thomas},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Dekking's conjecture; sum of digits; Newman phenomenon},
language = {eng},
month = {4},
number = {1},
pages = {17-24},
publisher = {Société Arithmétique de Bordeaux},
title = {On a conjecture of Dekking : The sum of digits of even numbers},
url = {http://eudml.org/doc/275820},
volume = {26},
year = {2014},
}

TY - JOUR
AU - Boreico, Iurie
AU - El-Baz, Daniel
AU - Stoll, Thomas
TI - On a conjecture of Dekking : The sum of digits of even numbers
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2014/4//
PB - Société Arithmétique de Bordeaux
VL - 26
IS - 1
SP - 17
EP - 24
AB - Let $q\ge 2$ and denote by $s_q$ the sum-of-digits function in base $q$. For $j=0,1,\dots ,q-1$ consider\[\# \lbrace 0 \le n &lt; N : \;\;s_q(2n) \equiv j \pmod {q} \rbrace .\]In 1983, F. M. Dekking conjectured that this quantity is greater than $N/q$ and, respectively, less than $N/q$ for infinitely many $N$, thereby claiming an absence of a drift (or Newman) phenomenon. In this paper we prove his conjecture.
LA - eng
KW - Dekking's conjecture; sum of digits; Newman phenomenon
UR - http://eudml.org/doc/275820
ER -

References

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