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Proceedings of the Steklov Institute of Mathematics

, Volume 299, Issue 1, pp 157–177 | Cite as

Haas Molnar Continued Fractions and Metric Diophantine Approximation

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Abstract

Haas–Molnar maps are a family of maps of the unit interval introduced by A. Haas and D. Molnar. They include the regular continued fraction map and A. Renyi’s backward continued fraction map as important special cases. As shown by Haas and Molnar, it is possible to extend the theory of metric diophantine approximation, already well developed for the Gauss continued fraction map, to the class of Haas–Molnar maps. In particular, for a real number x, if (p n /q n )n≥1 denotes its sequence of regular continued fraction convergents, set θ n (x) = q n 2 |xp n /q n |, n = 1, 2.... The metric behaviour of the Cesàro averages of the sequence (θ n (x))n≥1 has been studied by a number of authors. Haas and Molnar have extended this study to the analogues of the sequence (θ n (x))n≥1 for the Haas–Molnar family of continued fraction expansions. In this paper we extend the study of \(({\theta _{{k_n}}}(x))\)n≥1 for certain sequences (k n )n≥1, initiated by the second named author, to Haas–Molnar maps.

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References

  1. 1.
    G. Alkauskas, “Transfer operator for the Gauss’ continued fraction map. I: Structure of the eigenvalues and trace formulas,” arXiv: 1210.4083v6 [math.NT].Google Scholar
  2. 2.
    K. I. Babenko, “On a problem of Gauss,” Dokl. Akad. Nauk SSSR 238 (5), 1021–1024 (1978) [Sov. Math., Dokl. 19, 136–140 (1978)].MathSciNetMATHGoogle Scholar
  3. 3.
    K. I. Babenko and S. P. Jur’ev, “On the discretization of a problem of Gauss,” Dokl. Akad. Nauk SSSR 240 (6), 1273–1276 (1978) [Sov. Math., Dokl. 19, 731–735 (1978)].MathSciNetGoogle Scholar
  4. 4.
    A. Bellow, R. Jones, and J. Rosenblatt, “Convergence for moving averages,” Ergodic Theory Dyn. Syst. 10 (1), 43–62 (1990).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    M. Boshernitzan, G. Kolesnik, A. Quas, and M. Wierdl, “Ergodic averaging sequences,” J. Anal. Math. 95, 63–103 (2005).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    W. Bosma, H. Jager, and F. Wiedijk, “Some metrical observations on the approximation by continued fractions,” Indag. Math. 45, 281–299 (1983).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    J. Bourgain, “On the maximal ergodic theorem for certain subsets of the integers,” Isr. J. Math. 61 (1), 39–72 (1988).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    J. Bourgain (with an appendix jointly with H. Furstenberg, Y. Katznelson, and D. S. Ornstein), “Pointwise ergodic theorems for arithmetic sets,” Publ. Math., Inst. Hautes Étud. Sci. 69, 5–45 (1989).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    R. M. Burton, C. Kraaikamp, and T. A. Schmidt, “Natural extensions for the Rosen fractions,” Trans. Am. Math. Soc. 352 (3), 1277–1298 (2000).MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ, Ergodic Theory (Springer, New York, 1982), Grundl. Math. Wiss. 245.CrossRefMATHGoogle Scholar
  11. 11.
    K. Dajani, C. Kraaikamp, and N. van der Wekken, “Ergodicity of N-continued fraction expansions,” J. Number Theory 133 (9), 3183–3204 (2013).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    K. Gröchenig and A. Haas, “Backward continued fractions, Hecke groups and invariant measures for transformations of the interval,” Ergodic Theory Dyn. Syst. 16 (6), 1241–1274 (1996).MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    A. Haas, “Invariant measures and natural extensions,” Can. Math. Bull. 45 (1), 97–108 (2002).MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    A. Haas and D. Molnar, “Metrical diophantine approximation for continued fraction like maps of the interval,” Trans. Am. Math. Soc. 356 (7), 2851–2870 (2004).MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    A. Haas and D. Molnar, “The distribution of Jager pairs for continued fraction like mappings of the interval,” Pac. J. Math 217 (1), 101–114 (2004).MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    D. Hensley, “Continued fraction Cantor sets, Hausdorff dimension, and functional analysis,” J. Number Theory 40 (3), 336–358 (1992).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    D. Hensley, “Metric Diophantine approximation and probability,” New York J. Math. 4, 249–257 (1998).MathSciNetMATHGoogle Scholar
  18. 18.
    C. T. Ionescu Tulcea and G. Marinescu, “Théorie ergodique pour des classes d’opérations non complètement continues,” Ann. Math., Ser. 2, 52 (1), 140–147 (1950).CrossRefMATHGoogle Scholar
  19. 19.
    S. Lang, Introduction to Diophantine Approximations (Addison-Wesley, Reading, MA, 1966), Addison-Wesley Ser. Math.MATHGoogle Scholar
  20. 20.
    D. H. Mayer, The Ruelle–Araki Transfer Operator in Classical Statistical Mechanics (Springer, Berlin, 1980), Lect. Notes Phys. 123.MATHGoogle Scholar
  21. 21.
    D. Mayer and G. Roepstorff, “On the relaxation time of Gauss’s continued-fraction map. I: The Hilbert space approach (Koopmanism),” J. Stat. Phys. 47 (1–2), 149–171 (1987).MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    D. Mayer and G. Roepstorff, “On the relaxation time of Gauss’ continued-fraction map. II: The Banach space approach (transfer operator method),” J. Stat. Phys. 50 (1–2), 331–344 (1988).MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    R. Nair, “On polynomials in primes and J. Bourgain’s circle method approach to ergodic theorems. II,” Stud. Math. 105 (3), 207–233 (1993).MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    R. Nair, “On the metrical theory of continued fractions,” Proc. Am. Math. Soc. 120 (4), 1041–1046 (1994).MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    R. Nair, “On uniformly distributed sequences of integers and Poincaré recurrence. II,” Indag. Math., New Ser. 9 (3), 405–415 (1998).MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    R. Nair, “On metric Diophantine approximation and subsequence ergodic theory,” New York J. Math. 3A, 117–124 (1998).MathSciNetMATHGoogle Scholar
  27. 27.
    R. Nair and M. Weber, “On random perturbation of some intersective sets,” Indag. Math., New Ser. 15 (3), 373–381 (2004).MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    H. Nakada, “Metrical theory for a class of continued fraction transformations and their natural extensions,” Tokyo J. Math. 4, 399–426 (1981).MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    S. M. Rudolfer and K. M. Wilkinson, “A number-theoretic class of weak Bernoulli transformations,” Math. Syst. Theory 7, 14–24 (1973).MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    D. Ruelle, “Dynamical zeta functions and transfer operators,” Notices Am. Math. Soc. 49 (8), 887–895 (2002).MathSciNetMATHGoogle Scholar
  31. 31.
    D. Ruelle, Thermodynamic Formalism: The Mathematical Structures of Equilibrium Statistical Mechanics, 2nd ed. (Cambridge Univ. Press, Cambridge, 2004).CrossRefMATHGoogle Scholar
  32. 32.
    A. Tempelman, Ergodic Theorems for Group Actions: Informational and Thermodynamical Aspects (Kluwer, Dordrecht, 1992), Math. Appl. 78.CrossRefGoogle Scholar
  33. 33.
    L. Vepštas, “The Gauss–Kuzmin–Wirsing operator,” http://67.198.37.16/math/gkw.pdfGoogle Scholar
  34. 34.
    P. Walters, An Introduction to Ergodic Theory (Springer, New York, 1981).MATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Department of MathematicsBinzhou UniversityCity of Binzhou, Shandong ProvinceP.R. China
  2. 2.Department of Mathematical SciencesThe University of Liverpool, Mathematical Sciences BuildingLiverpoolUK

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