Geodesic
Spin chains and Arnold's problem on the Gauss-Kuz'min statistics for quadratic irrationals
Sbornik. Mathematics, Tome 204 (2013) no. 5, pp. 762-779 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

New results related to number theoretic model of spin chains are proved. We solve Arnold's problem on the Gauss-Kuz'min statistics for quadratic irrationals. Bibliography: 24 titles.
Keywords: continued fractions, Kloosterman sums, quadratic irrationals. 
@article{SM_2013_204_5_a5,
     author = {A. V. Ustinov},
     title = {Spin chains and {Arnold's} problem on the {Gauss-Kuz'min} statistics for quadratic irrationals},
     journal = {Sbornik. Mathematics},
     pages = {762--779},
     year = {2013},
     volume = {204},
     number = {5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2013_204_5_a5/}
}
TY  - JOUR
AU  - A. V. Ustinov
TI  - Spin chains and Arnold's problem on the Gauss-Kuz'min statistics for quadratic irrationals
JO  - Sbornik. Mathematics
PY  - 2013
SP  - 762
EP  - 779
VL  - 204
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/SM_2013_204_5_a5/
LA  - en
ID  - SM_2013_204_5_a5
ER  - 
%0 Journal Article
%A A. V. Ustinov
%T Spin chains and Arnold's problem on the Gauss-Kuz'min statistics for quadratic irrationals
%J Sbornik. Mathematics
%D 2013
%P 762-779
%V 204
%N 5
%U http://geodesic.mathdoc.fr/item/SM_2013_204_5_a5/
%G en
%F SM_2013_204_5_a5
A. V. Ustinov. Spin chains and Arnold's problem on the Gauss-Kuz'min statistics for quadratic irrationals. Sbornik. Mathematics, Tome 204 (2013) no. 5, pp. 762-779. http://geodesic.mathdoc.fr/item/SM_2013_204_5_a5/

[1] P. Kleban, A. E. Özlük, “A Farey fraction spin chain”, Comm. Math. Phys., 203:3 (1999), 635–647 | DOI | MR | Zbl

[2] J. Fiala, P. Kleban, A. E. Özlük, “The phase transition in statistical models defined on Farey fractions”, J. Statist. Phys., 110:1–2 (2003), 73–86 | DOI | MR | Zbl

[3] J. Kallies, A. Özlük, M. Peter, C. Snyder, “On asymptotic properties of a number theoretic function arising out of a spin chain model in statistical mechanics”, Comm. Math. Phys., 222:1 (2001), 9–43 | DOI | MR | Zbl

[4] M. Peter, “The limit distribution of a number theoretic function arising from a problem in statistical mechanics”, J. Number Theory, 90:2 (2001), 265–280 | DOI | MR | Zbl

[5] F. P. Boca, “Products of matrices $\bigl(\begin{smallmatrix} 1 1 \\\ 0 1 \end{smallmatrix}\bigr)$ and $\bigl(\begin{smallmatrix} 1 0\\\ 1 1 \end{smallmatrix}\bigr) $ and the distribution of reduced quadratic irrationals”, J. Reine Angew. Math., 606 (2007), 149–165 | DOI | MR | Zbl

[6] C. Faivre, “Distribution of Lévy constants for quadratic numbers”, Acta Arith., 61:1 (1992), 13–34 | MR | Zbl

[7] V. Arnold, Arnold's problems, Springer-Verlag, Berlin, 2004 | MR | MR | Zbl | Zbl

[8] A. V. Ustinov, “On the number of solutions of the congruence $xy\equiv l(\operatorname{mod}q)$ under the graph of a twice continuously differentiable function”, St. Petersburg Math. J., 20:5 (2009), 597–627 | DOI | MR

[9] T. Estermann, “On Kloosterman's sum”, Mathematika, 8 (1961), 83–86 | DOI | MR | Zbl

[10] V. A. Bykovskii, “Estimate for dispersion of lengths of continued fractions”, J. Math. Sci. (N. Y.), 146:2 (2007), 5634–5643 | DOI | MR | Zbl

[11] P. Lévy, “Sur les lois de probabilité dont dépendent les quotients complets et incomplets d'une fraction continue”, Bull. Soc. Math. France, 57 (1929), 178–194 | MR | Zbl

[12] G. Lochs, “Statistik der Teilnenner der zu den echten Brüchen gehörigen regelmässigen Kettenbrüche”, Monatsh. Math., 65 (1961), 27–52 | DOI | MR | Zbl

[13] M. O. Avdeeva, “On the statistics of partial quotients of finite continued fractions”, Funct. Anal. Appl., 38:2 (2004), 79–87 | DOI | DOI | MR | Zbl

[14] M. O. Avdeeva, V. A. Bykovskii, Reshenie zadachi Arnolda o statistikakh Gaussa–Kuzmina, Dalnauka, Vladivostok, 2002

[15] A. V. Ustinov, “On Gauss–Kuz'min statistics for finite continued fractions”, J. Math. Sci. (N. Y.), 146:2 (2007), 5771–5781 | DOI | MR | Zbl

[16] V. A. Bykovskii, A. V. Ustinov, “The statistics of particle trajectories in the inhomogeneous Sinai problem for a two-dimensional lattice”, Izv. Math., 73:4 (2009), 669–688 | DOI | DOI | MR | Zbl

[17] A. V. Ustinov, “The mean number of steps in the Euclidean algorithm with odd partial quotients”, Math. Notes, 88:4 (2010), 574–584 | DOI | DOI | MR | Zbl

[18] F. P. Boca, J. Vandehey, “On certain statistical properties of continued fractions with even and with odd partial quotients”, Acta Arith., 156:3 (2012), 201–221 | DOI | Zbl

[19] A. V. Ustinov, “On the distribution of Frobenius numbers with three arguments”, Izv. Math., 74:5 (2010), 1023–1049 | DOI | DOI | MR | Zbl

[20] I. M. Vinogradov, An introduction to the theory of numbers, Pergamon Press, London–New York, 1955 | MR | MR

[21] É. Galois, “Analyse algébrique. Démonstration d'un théorème sur les fractions continues périodiques”, Ann. Math. Pures Appl. [Ann. Gergonne], 19 (1828/29), 294–301 | MR

[22] P. Sarnak, “Class numbers of indefinite binary quadratic forms”, J. Number Theory, 15:2 (1982), 229–247 | DOI | MR | Zbl

[23] H. J. S. Smith, “Note on the theory of the Pellian equation and of binary quadratic forms of a positive determinant”, Proc. London Math. Soc., s1-7 (1875), 196–208 | DOI | MR

[24] B. A. Venkov, Elementary number theory, Wolters-Noordhoff Publ., Groningen, 1970 | MR | Zbl