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 Chebyshevskii Sb., 2018, Volume 19, Issue 2, Pages 501–522 (Mi cheb669)

Substitutions and bounded remainder sets

A. V. Shutov

Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: The paper is devoted to the multidimensional problem of distribution of fractional parts of a linear function. A subset of a multidimensional torus is called a bounded remainder set if the remainder term of the multidimensional problem of the distribution of the fractional parts of a linear function on this set is bounded by an absolute constant. We are interested not only in the individual bounded remainder sets but also in toric tilings into such sets.
A new class of tilings of a $d$-dimensional torus into sets of $(d + 1)$ types is introduced. These tilings are defined in combinatorics and geometric terms and are called generalized exchanged tilings. It is proved that all generalized exchanged toric tilings consist of bounded remainder sets. Corresponding estimate of the remainder term is effective. We also find conditions that ensure that the estimate of the remainder term for the sequence of generalized exchanged toric tilings does not depend on the concrete tiling in the sequence.
Using the Arnoux-Ito theory of geometric substitutions we introduce a new class of generalized exchanged tilings of multidimensional tori into bounded remainder sets with an effective estimate of the remainder term. Earlier similar results were obtained in the two-dimensional case for one specific substitution — a geometric version of well-known Rauzy substitution. With the help of the passage to the limit, another class of generalized exchanged toric tilings into bounded remainder sets with fractal boundaries is constructed (so-called generalized Rauzy fractals).

Keywords: uniform distribution, bounded remainder sets, toric tilings, unimodular Pisot substitutions, geometric substitutions.

 Funding Agency Grant Number Russian Science Foundation 14-11-00433

DOI: https://doi.org/10.22405/2226-8383-2018-19-2-501-522

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Bibliographic databases:

UDC: 511.43
Accepted:17.08.2018

Citation: A. V. Shutov, “Substitutions and bounded remainder sets”, Chebyshevskii Sb., 19:2 (2018), 501–522

Citation in format AMSBIB
\Bibitem{Shu18} \by A.~V.~Shutov \paper Substitutions and bounded remainder sets \jour Chebyshevskii Sb. \yr 2018 \vol 19 \issue 2 \pages 501--522 \mathnet{http://mi.mathnet.ru/cheb669} \crossref{https://doi.org/10.22405/2226-8383-2018-19-2-501-522} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3772444} \elib{https://elibrary.ru/item.asp?id=37112169} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. A. A. Zhukova, A. V. Shutov, “$n$-korony v razbieniyakh tora na mnozhestva ogranichennogo ostatka”, Chebyshevskii sb., 20:3 (2019), 246–260
2. A. V. Shutov, “Obobschennye razbieniya Rozi i mnozhestva ogranichennogo ostatka”, Chebyshevskii sb., 20:3 (2019), 372–389
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