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 Uspekhi Mat. Nauk, 2003, Volume 58, Issue 1(349), Pages 113–164 (Mi umn594)

Generalized continued fractions and ergodic theory

L. D. Pustyl'nikov

M. V. Keldysh Institute for Applied Mathematics, Russian Academy of Sciences

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.

DOI: https://doi.org/10.4213/rm594

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English version:
Russian Mathematical Surveys, 2003, 58:1, 109–159

Bibliographic databases:

UDC: 511.335+511.336+517.987.5
MSC: Primary 11J70, 28D05; Secondary 11A55, 11K50, 30B70, 11L15, 11J54, 37A05

Citation: L. D. Pustyl'nikov, “Generalized continued fractions and ergodic theory”, Uspekhi Mat. Nauk, 58:1(349) (2003), 113–164; Russian Math. Surveys, 58:1 (2003), 109–159

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/umn594
• https://doi.org/10.4213/rm594
• http://mi.mathnet.ru/eng/umn/v58/i1/p113

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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. Georgiev G.H., Glazunov N.M., Krakovsky V.Y., Kumkov S.I., Noel A.G., Pustyl'nikov L.D., Wicks M.C., Himed B., “Selected problems”, Computational Noncommutative Algebra and Applications, Nato Science Series, Series II: Mathematics, Physics and Chemistry, 136, 2004, 413–424
2. V. N. Berestovskii, Yu. G. Nikonorov, “Continued Fractions, the Group $\mathrm{GL}(2,\mathbb Z)$, and Pisot Numbers”, Siberian Adv. Math., 17:4 (2007), 268–290
3. Schratzberger B., “On the singularization of the two-dimensional Jacobi-Perron algorithm”, Experiment. Math., 16:4 (2007), 441–454
4. L. D. Pustylnikov, T. V. Lokot, “Diskretnye povoroty i obobschënnye tsepnye drobi”, Preprinty IPM im. M. V. Keldysha, 2009, 044, 7 pp.
5. A. D. Bryuno, “Universalnoe obobschenie algoritma tsepnoi drobi”, Chebyshevskii sb., 16:2 (2015), 35–65
6. V. G. Zhuravlev, “Simplex-module algorithm for expansion of algebraic numbers in multidimensional continued fractions”, J. Math. Sci. (N. Y.), 225:6 (2017), 924–949
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