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Zap. Nauchn. Sem. POMI, 2016, Volume 445, Pages 93–174 (Mi znsl6276)  

This article is cited in 3 scientific papers (total in 3 papers)

Bounded remainder sets

V. G. Zhuravlev

Vladimir State University, Vladimir, Russia

Abstract: We consider the category $(\mathcal{T,S,X})$ consisting of transformations $\mathcal{S\colon T\to T}$ of spaces $\mathcal T$ with distinguished subsets $\mathcal{X\subset T}$. Let $r_\mathcal X(i,x_0)$ be the distribution function of points from the $\mathcal S$-orbit $x_0,x_1=\mathcal S(x_0),…,x_{i-1}=\mathcal S^{i-1}(x_0)$ got in $\mathcal X$, and a deviation $\delta_\mathcal X(i,x_0)$ be defined by the equation
$$ r_\mathcal X(i,x_0)=a_\mathcal Xi+\delta_\mathcal X(i,x_0), $$
where $a_\mathcal Xi$ is the average value. If $\delta_\mathcal X(i,x_0)=O(1)$ then such $\mathcal X$ are called bounded remainder sets. In this article the bounded remainder sets $\mathcal X$ are built in the following cases: 1) the space $\mathcal T$ is a circle, a torus or a Klein bottle; 2) the map $\mathcal S$ is a rotation of the circle, a shift or an exchange transformation of the torus; 3) the $\mathcal X$ is a fixed subset $\mathcal{X\subset T}$ or a sequence of subsets depending on the iteration step $i=0,1,2,…$

Key words and phrases: toric exchange, induced decomposition, bounded remainder sets.

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00360


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English version:
Journal of Mathematical Sciences (New York), 2017, 222:5, 585–640

Bibliographic databases:

UDC: 511
Received: 16.01.2016

Citation: V. G. Zhuravlev, “Bounded remainder sets”, Analytical theory of numbers and theory of functions. Part 31, Zap. Nauchn. Sem. POMI, 445, POMI, St. Petersburg, 2016, 93–174; J. Math. Sci. (N. Y.), 222:5 (2017), 585–640

Citation in format AMSBIB
\Bibitem{Zhu16}
\by V.~G.~Zhuravlev
\paper Bounded remainder sets
\inbook Analytical theory of numbers and theory of functions. Part~31
\serial Zap. Nauchn. Sem. POMI
\yr 2016
\vol 445
\pages 93--174
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6276}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3511160}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2017
\vol 222
\issue 5
\pages 585--640
\crossref{https://doi.org/10.1007/s10958-017-3322-7}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85015706835}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. G. Zhuravlev, “Differentiation of induced toric tilings and multi-dimensional approximations of algebraic numbers”, J. Math. Sci. (N. Y.), 222:5 (2017), 544–584  mathnet  crossref  mathscinet
    2. A. V. Shutov, “Nonautonomous bounded remainder sets”, Russian Mathematics, 62:12 (2018), 81–87  mathnet  crossref  isi
    3. V. G. Zhuravlev, “Unimodulyarnost indutsirovannykh razbienii tora”, Algebra i teoriya chisel. 1, Posvyaschaetsya pamyati Olega Mstislavovicha FOMENKO, Zap. nauchn. sem. POMI, 469, POMI, SPb., 2018, 64–95  mathnet
  • Записки научных семинаров ПОМИ
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