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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

On the asymptotic behaviour of the Lebesgue measure of sum-level sets for continued fractions

Pages: 2437 - 2451, Volume 32, Issue 7, July 2012      doi:10.3934/dcds.2012.32.2437

 
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Marc Kessböhmer - Universität Bremen, Fachbereich 3 - Mathematik und Informatik, Bibliothekstr. 1, 28359 Bremen, Germany (email)
Bernd O. Stratmann - Universität Bremen, Fachbereich 3 - Mathematik und Informatik, Bibliothekstr. 1, 28359 Bremen, Germany (email)

Abstract: In this paper we give a detailed measure theoretical analysis of what we call sum-level sets for regular continued fraction expansions. The first main result is to settle a recent conjecture of Fiala and Kleban, which asserts that the Lebesgue measure of these level sets decays to zero, for the level tending to infinity. The second and third main results then give precise asymptotic estimates for this decay. The proofs of these results are based on recent progress in infinite ergodic theory, and in particular, they give non-trivial applications of this theory to number theory. The paper closes with a discussion of the thermodynamical significance of the obtained results, and with some applications of these to metrical Diophantine analysis.

Keywords:  Continued fractions, thermodynamical formalism, multifractals, infinite ergodic theory, phase transition, intermittency, Stern-Brocot sequence, Farey sequence, Gauss map, Farey map.
Mathematics Subject Classification:  Primary: 37A45; Secondary: 11J70, 11J83, 28A80, 20H10.

Received: December 2009;      Revised: June 2010;      Available Online: March 2012.

 References