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On Nearly Linear Recurrence Sequences Read chapter Read first chapter

2017 | OriginalPaper | Chapter

Sums of the Digits in Bases 2 and 3

Authors : Jean-Marc Deshouillers, Laurent Habsieger, Shanta Laishram, Bernard Landreau

Published in: Number Theory – Diophantine Problems, Uniform Distribution and Applications

Publisher: Springer International Publishing

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Abstract

Let b ≥ 2 be an integer and let s b (n) denote the sum of the digits of the representation of an integer n in base b. For sufficiently large N, one has
$$\displaystyle{\mathop{\mathrm{Card}}\nolimits \{n \leq N: \left \vert s_{3}(n) - s_{2}(n)\right \vert \leq 0.1457205\log n\}\,>\, N^{0.970359}.}$$
The proof only uses the separate (or marginal) distributions of the values of s 2(n) and s 3(n).

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previous chapter Local Oscillations in Moderately Dense Sequences of Primes
next chapter On the Discrepancy of Halton–Kronecker Sequences
Footnotes
1
Indeed, if the last non-zero digit of n! in base 12 belongs to {1, 2, 5, 7, 10, 11} then | s 3(n) − s 2(n) | ≤ 1; this seems to occur infinitely many times.
 
2
For example, when L = 5, the number “sixty” will be represented as 02020. Happy palindromic birthday, Robert!
 
Literature
1.
N.L. Bassily, I. Kátai, Distribution of the values of q-additive functions, on polynomial sequences. Acta Math. Hungar. 68, 353–361 (1995)MathSciNetCrossRefMATH
2.
J.-M. Deshouillers, A footnote to The least non zero digit of n! in base 12. Unif. Distrib. Theory 7, 71–73 (2012)
3.
J.-M. Deshouillers, I. Ruzsa, The least non zero digit of n! in base 12. Publ. Math. Debr. 79, 395–400 (2011)CrossRefMATH
4.
5.
C. Stewart, On the representation of an integer in two different bases. J. Reine Angew. Math. 319, 63–72 (1980)MathSciNetMATH
Metadata
Title
Sums of the Digits in Bases 2 and 3
Authors
Jean-Marc Deshouillers
Laurent Habsieger
Shanta Laishram
Bernard Landreau
Copyright Year
2017
Book
Number Theory – Diophantine Problems, Uniform Distribution and Applications

Print ISBN: 978-3-319-55356-6

Electronic ISBN: 978-3-319-55357-3

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-319-55357-3_9

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