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Russian Mathematical Surveys, 2003, Volume 58, Issue 1, Pages 109–159
DOI: https://doi.org/10.1070/RM2003v058n01ABEH000594
(Mi rm594)
 

This article is cited in 7 scientific papers (total in 7 papers)

Generalized continued fractions and ergodic theory

L. D. Pustyl'nikov

M. V. Keldysh Institute for Applied Mathematics, Russian Academy of Sciences
References:
Abstract: In this paper a new theory of generalized continued fractions is constructed and applied to numbers, multidimensional vectors belonging to a real space, and infinite-dimensional vectors with integral coordinates. The theory is based on a concept generalizing the procedure for constructing the classical continued fractions and substantially using ergodic theory. One of the versions of the theory is related to differential equations. In the finite-dimensional case the constructions thus introduced are used to solve problems posed by Weyl in analysis and number theory concerning estimates of trigonometric sums and of the remainder in the distribution law for the fractional parts of the values of a polynomial, and also the problem of characterizing algebraic and transcendental numbers with the use of generalized continued fractions. Infinite-dimensional generalized continued fractions are applied to estimate sums of Legendre symbols and to obtain new results in the classical problem of the distribution of quadratic residues and non-residues modulo a prime. In the course of constructing these continued fractions, an investigation is carried out of the ergodic properties of a class of infinite-dimensional dynamical systems which are also of independent interest.
Received: 05.01.2000
Bibliographic databases:
Document Type: Article
UDC: 511.335+511.336+517.987.5
MSC: Primary 11J70, 28D05; Secondary 11A55, 11K50, 30B70, 11L15, 11J54, 37A05
Language: English
Original paper language: Russian
Citation: L. D. Pustyl'nikov, “Generalized continued fractions and ergodic theory”, Russian Math. Surveys, 58:1 (2003), 109–159
Citation in format AMSBIB
\Bibitem{Pus03}
\by L.~D.~Pustyl'nikov
\paper Generalized continued fractions and ergodic theory
\jour Russian Math. Surveys
\yr 2003
\vol 58
\issue 1
\pages 109--159
\mathnet{http://mi.mathnet.ru//eng/rm594}
\crossref{https://doi.org/10.1070/RM2003v058n01ABEH000594}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1992132}
\zmath{https://zbmath.org/?q=an:1063.37007}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2003RuMaS..58..109P}
\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=000183858300003}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-30244545661}
Linking options:
  • https://www.mathnet.ru/eng/rm594
  • https://doi.org/10.1070/RM2003v058n01ABEH000594
  • https://www.mathnet.ru/eng/rm/v58/i1/p113
  • This publication is cited in the following 7 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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