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Chebyshevskii Sbornik, 2021, Volume 22, Issue 2, Pages 104–120
DOI: https://doi.org/10.22405/2226-8383-2018-22-2-104-120
(Mi cheb1025)
 

This article is cited in 2 scientific papers (total in 2 papers)

On Gelfond-type problem for generalized Zeckendorf representations

A. A. Zhukovaa, A. V. Shutovb

a Russian Academy of National Economy and Public Administration under the President of Russian Federation, Vladimir branch (Vladimir)
b Vladimir State University named after Alexander and Nicholay Stoletovs (Vladimir)
Full-text PDF (838 kB) Citations (2)
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Abstract: Gelfond proved that for coprime $b-1$ and $d$ sums of digits of $b$-ary expressions of natural numbers are uniformly distributed on arithmetic progressions with the common difference $d$. Later, similar result was proved for the representations of natural numbers based on linear recurrent sequences.
We consider the remainder term of the corresponding asymptotic formula and study the dichotomy between the logarithmic and power estimates of the remainder term. In the case $d=2$, we obtain some sufficient condition for the validity of the logarithmic estimate. Using them, we show that the logarithmic estimate holds for expansions based on all second-order linear recurrenct sequences and on infinite family of third-order sequences. Also we construct an example of the linear reccurent sequence of an arbitrary order with such property. On the other hand, we give an example of a third-order linear recurrent sequence for which the logarithmic estimate does not hold. We also show that for $d=3$ the logarithmic estimate does not hold even in the simplest case of the expansions based on Fibonacci numbers.
In addition, we consider the representations of natural numbers based on the denominators of partial convergents of the continued fraction expansions of irrational numbers. In this case, we prove the uniformity of the distribution of sums of digits over arithmetic progressions with the common difference $2$ with the logarithmic remainder term.
Keywords: Fibonacci numers, generalized Zeckendorf representations, linear reccurent sequences, continued fractions, sums of digints, Gelfond problem.
Document Type: Article
UDC: 517
Language: Russian
Citation: A. A. Zhukova, A. V. Shutov, “On Gelfond-type problem for generalized Zeckendorf representations”, Chebyshevskii Sb., 22:2 (2021), 104–120
Citation in format AMSBIB
\Bibitem{ZhuShu21}
\by A.~A.~Zhukova, A.~V.~Shutov
\paper On Gelfond-type problem for generalized Zeckendorf representations
\jour Chebyshevskii Sb.
\yr 2021
\vol 22
\issue 2
\pages 104--120
\mathnet{http://mi.mathnet.ru/cheb1025}
\crossref{https://doi.org/10.22405/2226-8383-2018-22-2-104-120}
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  • https://www.mathnet.ru/eng/cheb/v22/i2/p104
  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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    References:10
     
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