Large deviations for denominators of continued fractions

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Abstract

We give exponential upper bounds on the probability with which the denominator of the nth convergent in the regular continued fraction expansion stays away from the mean 12nlogπ22. The exponential rate is best possible, given by an analytic function related to the dimension spectrum of Lyapunov exponents for the Gauss transformation. We also establish the large deviation principle (LDP) for denominators. As corollaries, we derive the LDPs for denominators of periodic continued fractions and continued fraction preimages. Proofs of the main results rely on the thermodynamic formalism for finite topological Markov shifts.

Original languageEnglish
Pages (from-to)5861-5874
Number of pages14
JournalNonlinearity
Volume33
Issue number11
DOIs
Publication statusPublished - 2020 Nov

Keywords

  • Continued fraction
  • Large deviation principle
  • Multifractal analysis
  • Thermodynamic formalism

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

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