2012, 32(7): 2437-2451. doi: 10.3934/dcds.2012.32.2437

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

1. 

Universität Bremen, Fachbereich 3 - Mathematik und Informatik, Bibliothekstr. 1, 28359 Bremen, Germany, Germany

Received  December 2009 Revised  June 2010 Published  March 2012

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.
Citation: Marc Kessböhmer, Bernd O. Stratmann. On the asymptotic behaviour of the Lebesgue measure of sum-level sets for continued fractions. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2437-2451. doi: 10.3934/dcds.2012.32.2437
References:
[1]

J. Aaronson, "An Introduction to Infinite Ergodic Theory," Mathematical Surveys and Monographs, 50,, American Mathematical Society, (1997).

[2]

J. Aaronson, M. Denker and M. Urbański, Ergodic theory for Markov fibred systems and parabolic rational maps,, Trans. AMS, 337 (1993), 495. doi: 10.2307/2154231.

[3]

N. H. Bingham, C. M. Goldie and J. L. Teugels, "Regular Variation," Encyclopedia of Mathematics and its Applications, 27,, Cambridge University Press, (1989).

[4]

A. Brocot, Calcul des rouages par approximation, nouvelle méthode,, Revue chronométrique, 3 (1981), 186.

[5]

H. Bruin, M. Nicol and D. Terhesiu, On Young towers associated with infinite measure preserving transformations,, Stoch. and Dynamics, 9 (2009), 635. doi: 10.1142/S0219493709002816.

[6]

H. E. Daniels, Processes generating permutation expansions,, Biometrika, 49 (1962), 139. doi: 10.1093/biomet/49.1-2.139.

[7]

J. L. Doob, "Stochastic Processes,", John Wiley & Sons, (1953).

[8]

M. J. Feigenbaum, I. Procaccia and T. Tél, Scaling properties of multifractals as an eigenvalue problem,, Phys. Rev. A (3), 39 (1989), 5359. doi: 10.1103/PhysRevA.39.5359.

[9]

J. Fiala and P. Kleban, Intervals between Farey fractions in the limit of infinite level,, Annales des Sciences Mathematiques du Québec, 34 (2010), 63.

[10]

D. Hensley, The statistics of the continued fraction digit sum,, Pacific Jour. of Math., 192 (2000), 103. doi: 10.2140/pjm.2000.192.103.

[11]

B. Hu and J. Rudnik, Exact solutions to the Feigenbaum renormalization-group equations for intermittency,, Phys. Rev. Lett., 48 (1982), 1645.

[12]

M. Kesseböhmer, S. Munday and B. O. Stratmann, Strong renewal theorems and Lyapunov spectra for $\alpha$-Farey and $\alpha$-Lüroth systems,, to appear in Ergod. Theory and Dyn. Syst., ().

[13]

M. Kesseböhmer and M. Slassi, Limit laws for distorted critical return time processes in infinite ergodic theory,, Stochastics and Dynamics, 7 (2007), 103.

[14]

M. Kesseböhmer and M. Slassi, A distributional limit law for the continued fraction digit sum,, Mathematische Nachrichten, 281 (2008), 1294.

[15]

M. Kesseböhmer and M. Slassi, Large deviation asymptotics for continued fraction expansions,, Stochastics and Dynamics, 8 (2008), 103.

[16]

M. Kesseböhmer and B. O. Stratmann, A multifractal formalism for growth rates and applications to geometrically finite Kleinian groups,, Ergodic Theory & Dynamical Systems, 24 (2004), 141.

[17]

M. Kesseböhmer and B. O. Stratmann, Stern-Brocot pressure and multifractal spectra in ergodic theory of numbers,, Stochastics and Dynamics, 4 (2004), 77. doi: 10.1142/S0219493704000948.

[18]

M. Kesseböhmer and B. O. Stratmann, A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates,, J. Reine Angew. Math., 605 (2007), 133.

[19]

A. Ya. Khintchine, "Continued Fractions,", Univ. of Chicago Press, (1964).

[20]

M. Lin, Mixing for Markov operators,, Z. Wahrsch. u. V. Geb., 19 (1971), 231. doi: 10.1007/BF00534111.

[21]

W. Parry, On $\beta$-expansions of real numbers,, Acta Math. Acad. Sci. Hung., 11 (1960), 401. doi: 10.1007/BF02020954.

[22]

W. Parry, Ergodic properties of some permutation processes,, Biometrika, 49 (1962), 151. doi: 10.2307/2333475.

[23]

T. Prellberg and J. Slawny, Maps of intervals with indifferent fixed points: Thermodynamical formalism and phase transition,, J. Stat. Phys., 66 (1992), 503.

[24]

M. A. Stern, Über eine zahlentheoretische Funktion,, J. Reine Angew. Math., 55 (1958), 193.

[25]

M. Thaler, Estimates of the invariant densities of endomorphisms with indifferent fixed points,, Israel J. Math., 37 (1980), 303.

[26]

M. Thaler, Transformations on $[0,1]$ with infinite invariant measures,, Israel J. Math., 46 (1983), 67. doi: 10.1007/BF02760623.

[27]

M. Thaler, The asymptotics of the Perron-Frobenius operator of a class of interval maps preserving infinite measures,, Studia Math., 143 (2000), 103.

[28]

M. Thaler, "Infinite Ergodic Theory,", Luminy lecture notes, (2001).

[29]

E. Wirsing, On the theorem of Gauss-Kusmin-Lévy and a Frobenius-type theorem for function spaces,, V. Acta Arith., 24 (): 507.

show all references

References:
[1]

J. Aaronson, "An Introduction to Infinite Ergodic Theory," Mathematical Surveys and Monographs, 50,, American Mathematical Society, (1997).

[2]

J. Aaronson, M. Denker and M. Urbański, Ergodic theory for Markov fibred systems and parabolic rational maps,, Trans. AMS, 337 (1993), 495. doi: 10.2307/2154231.

[3]

N. H. Bingham, C. M. Goldie and J. L. Teugels, "Regular Variation," Encyclopedia of Mathematics and its Applications, 27,, Cambridge University Press, (1989).

[4]

A. Brocot, Calcul des rouages par approximation, nouvelle méthode,, Revue chronométrique, 3 (1981), 186.

[5]

H. Bruin, M. Nicol and D. Terhesiu, On Young towers associated with infinite measure preserving transformations,, Stoch. and Dynamics, 9 (2009), 635. doi: 10.1142/S0219493709002816.

[6]

H. E. Daniels, Processes generating permutation expansions,, Biometrika, 49 (1962), 139. doi: 10.1093/biomet/49.1-2.139.

[7]

J. L. Doob, "Stochastic Processes,", John Wiley & Sons, (1953).

[8]

M. J. Feigenbaum, I. Procaccia and T. Tél, Scaling properties of multifractals as an eigenvalue problem,, Phys. Rev. A (3), 39 (1989), 5359. doi: 10.1103/PhysRevA.39.5359.

[9]

J. Fiala and P. Kleban, Intervals between Farey fractions in the limit of infinite level,, Annales des Sciences Mathematiques du Québec, 34 (2010), 63.

[10]

D. Hensley, The statistics of the continued fraction digit sum,, Pacific Jour. of Math., 192 (2000), 103. doi: 10.2140/pjm.2000.192.103.

[11]

B. Hu and J. Rudnik, Exact solutions to the Feigenbaum renormalization-group equations for intermittency,, Phys. Rev. Lett., 48 (1982), 1645.

[12]

M. Kesseböhmer, S. Munday and B. O. Stratmann, Strong renewal theorems and Lyapunov spectra for $\alpha$-Farey and $\alpha$-Lüroth systems,, to appear in Ergod. Theory and Dyn. Syst., ().

[13]

M. Kesseböhmer and M. Slassi, Limit laws for distorted critical return time processes in infinite ergodic theory,, Stochastics and Dynamics, 7 (2007), 103.

[14]

M. Kesseböhmer and M. Slassi, A distributional limit law for the continued fraction digit sum,, Mathematische Nachrichten, 281 (2008), 1294.

[15]

M. Kesseböhmer and M. Slassi, Large deviation asymptotics for continued fraction expansions,, Stochastics and Dynamics, 8 (2008), 103.

[16]

M. Kesseböhmer and B. O. Stratmann, A multifractal formalism for growth rates and applications to geometrically finite Kleinian groups,, Ergodic Theory & Dynamical Systems, 24 (2004), 141.

[17]

M. Kesseböhmer and B. O. Stratmann, Stern-Brocot pressure and multifractal spectra in ergodic theory of numbers,, Stochastics and Dynamics, 4 (2004), 77. doi: 10.1142/S0219493704000948.

[18]

M. Kesseböhmer and B. O. Stratmann, A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates,, J. Reine Angew. Math., 605 (2007), 133.

[19]

A. Ya. Khintchine, "Continued Fractions,", Univ. of Chicago Press, (1964).

[20]

M. Lin, Mixing for Markov operators,, Z. Wahrsch. u. V. Geb., 19 (1971), 231. doi: 10.1007/BF00534111.

[21]

W. Parry, On $\beta$-expansions of real numbers,, Acta Math. Acad. Sci. Hung., 11 (1960), 401. doi: 10.1007/BF02020954.

[22]

W. Parry, Ergodic properties of some permutation processes,, Biometrika, 49 (1962), 151. doi: 10.2307/2333475.

[23]

T. Prellberg and J. Slawny, Maps of intervals with indifferent fixed points: Thermodynamical formalism and phase transition,, J. Stat. Phys., 66 (1992), 503.

[24]

M. A. Stern, Über eine zahlentheoretische Funktion,, J. Reine Angew. Math., 55 (1958), 193.

[25]

M. Thaler, Estimates of the invariant densities of endomorphisms with indifferent fixed points,, Israel J. Math., 37 (1980), 303.

[26]

M. Thaler, Transformations on $[0,1]$ with infinite invariant measures,, Israel J. Math., 46 (1983), 67. doi: 10.1007/BF02760623.

[27]

M. Thaler, The asymptotics of the Perron-Frobenius operator of a class of interval maps preserving infinite measures,, Studia Math., 143 (2000), 103.

[28]

M. Thaler, "Infinite Ergodic Theory,", Luminy lecture notes, (2001).

[29]

E. Wirsing, On the theorem of Gauss-Kusmin-Lévy and a Frobenius-type theorem for function spaces,, V. Acta Arith., 24 (): 507.

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