
This article is cited in 2 scientific papers (total in 2 papers)
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 ArnouxIto 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 twodimensional case for one specific
substitution — a geometric version of wellknown 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 (socalled
generalized Rauzy fractals).
Keywords:
uniform distribution, bounded remainder sets, toric tilings, unimodular Pisot substitutions, geometric substitutions.
DOI:
https://doi.org/10.22405/222683832018192501522
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UDC:
511.43 Received: 10.06.2018 Accepted:17.08.2018
Citation:
A. V. Shutov, “Substitutions and bounded remainder sets”, Chebyshevskii Sb., 19:2 (2018), 501–522
Citation in format AMSBIB
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\by A.~V.~Shutov
\paper Substitutions and bounded remainder sets
\jour Chebyshevskii Sb.
\yr 2018
\vol 19
\issue 2
\pages 501522
\mathnet{http://mi.mathnet.ru/cheb669}
\crossref{https://doi.org/10.22405/222683832018192501522}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=3772444}
\elib{https://elibrary.ru/item.asp?id=37112169}
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This publication is cited in the following articles:

A. A. Zhukova, A. V. Shutov, “$n$korony v razbieniyakh tora na mnozhestva ogranichennogo ostatka”, Chebyshevskii sb., 20:3 (2019), 246–260

A. V. Shutov, “Obobschennye razbieniya Rozi i mnozhestva ogranichennogo ostatka”, Chebyshevskii sb., 20:3 (2019), 372–389

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