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Spin chains and Arnold’s problem on the Gauss-Kuz’min statistics for quadratic irrationals. (English. Russian original) Zbl 1288.11092
Sb. Math. 204, No. 5, 762-779 (2013); translation from Mat. Sb. 204, No. 5, 143-160 (2013).
A new asymptotic formula is obtained for the number of solutions of the congruence $$xy\equiv \pm 1 \pmod q$$ under the graph of linear function. The proof is based on the Weil bounds for Kloosterman sums. Using this result and the number theoretic model of spin chains, the author proves the asymptotic formula for the number of the chains, with the energy being bounded by a given number $$N$$. Moreover, the more general result concerning the Gauss-Kuzmin statistics for spin chains is proved in the paper. Using the relation between the considered characteristics of spin chains and the distribution of quadratic irrationals, the author solves the Arnold’s problem [V. I. Arnol’d, Arnold’s problems. (Russian), Moscow: FAZIS (2000; Zbl 1052.00003)] on the Gauss-Kuzmin statistics for quadratic irrationals.

##### MSC:
 11N37 Asymptotic results on arithmetic functions 11L07 Estimates on exponential sums 11K50 Metric theory of continued fractions 11A55 Continued fractions
##### Keywords:
continued fraction; Kloosterman sum; quadratic irrational
Zbl 1052.00003
Full Text:
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