Haas-Molnar continued fractions and metric Diophantine approximation. (English) Zbl 1397.37043
Proc. Steklov Inst. Math. 299, 157-177 (2017); translation in Tr. Mat. Inst. Steklova 299, 170-191 (2017).
For \(x \in [0,1]\), a well-known quantity \(\theta_n(x)\) measures the distance between \(x\) and the \(n\)-th convergent of its continued fraction expansion. A. Haas and D. Molnar [Trans. Am. Math. Soc. 356, No. 7, 2851–2870 (2004; Zbl 1051.11038)] introduced a family of piecewise Möbius transformations \(T_u:[0,1] \rightarrow [0,1]\) which includes the Gauss map. Those authors studied analogues of \((\theta_n(x))_{n \geq 1}\) for \(T_u\). The paper reviewed here extends results of R. Nair [New York J. Math. 3A, 117–124 (1998; Zbl 0894.11032)] from the Gauss map to \(T_u\), studying \((\theta_{k_n}(x))_{n \geq 1}\) for certain sequences \((k_n)_{n \geq 1}\). The authors’ results also generalize theorems of D. Hensley [New York J. Math. 4, 249–257 (1998; Zbl 0991.11046)] and A. Haas and D. Molnar [Pac. J. Math. 217, No. 1, 101–114 (2004; Zbl 1080.11060)].
Reviewer: Steve Pederson (Atlanta)
MSC:
| 37E05 | Dynamical systems involving maps of the interval |
| 11A55 | Continued fractions |
| 11J54 | Small fractional parts of polynomials and generalizations |
References:
| [1] | G. Alkauskas, “Transfer operator for the Gauss’ continued fraction map. I: Structure of the eigenvalues and trace formulas,” arXiv: 1210.4083v6 [math.NT]. |
| [2] | Babenko, K. I., On a problem of Gauss, Dokl. Akad. Nauk SSSR, 238, 1021-1024, (1978) · Zbl 0885.41020 |
| [3] | Babenko, K. I.; Jur’ev, S. P., On the discretization of a problem of Gauss, Dokl. Akad. Nauk SSSR, 240, 1273-1276, (1978) · Zbl 0416.10040 |
| [4] | Bellow, A.; Jones, R.; Rosenblatt, J., Convergence for moving averages, Ergodic Theory Dyn. Syst., 10, 43-62, (1990) · Zbl 0674.60035 · doi:10.1017/S0143385700005381 |
| [5] | Boshernitzan, M.; Kolesnik, G.; Quas, A.; Wierdl, M., Ergodic averaging sequences, J. Anal. Math., 95, 63-103, (2005) · Zbl 1100.28010 · doi:10.1007/BF02791497 |
| [6] | Bosma, W.; Jager, H.; Wiedijk, F., Some metrical observations on the approximation by continued fractions, Indag. Math., 45, 281-299, (1983) · Zbl 0519.10043 · doi:10.1016/1385-7258(83)90063-X |
| [7] | Bourgain, J., On the maximal ergodic theorem for certain subsets of the integers, Isr. J. Math., 61, 39-72, (1988) · Zbl 0642.28010 · doi:10.1007/BF02776301 |
| [8] | Bourgain, J., Pointwise ergodic theorems for arithmetic sets, Publ. Math., Inst. Hautes Étud. Sci., 69, 5-45, (1989) · Zbl 0705.28008 · doi:10.1007/BF02698838 |
| [9] | Burton, R. M.; Kraaikamp, C.; Schmidt, T. A., Natural extensions for the Rosen fractions, Trans. Am. Math. Soc., 352, 1277-1298, (2000) · Zbl 0938.11036 · doi:10.1090/S0002-9947-99-02442-3 |
| [10] | I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ, Ergodic Theory (Springer, New York, 1982), Grundl. Math. Wiss. 245. · Zbl 0493.28007 · doi:10.1007/978-1-4615-6927-5 |
| [11] | Dajani, K.; Kraaikamp, C.; Wekken, N., Ergodicity of \(N\)-continued fraction expansions, J. Number Theory, 133, 3183-3204, (2013) · Zbl 1325.11066 · doi:10.1016/j.jnt.2013.02.017 |
| [12] | Gröchenig, K.; Haas, A., Backward continued fractions, Hecke groups and invariant measures for transformations of the interval, Ergodic Theory Dyn. Syst., 16, 1241-1274, (1996) · Zbl 0884.58040 · doi:10.1017/S0143385700010014 |
| [13] | Haas, A., Invariant measures and natural extensions, Can. Math. Bull., 45, 97-108, (2002) · Zbl 1044.37002 · doi:10.4153/CMB-2002-011-4 |
| [14] | Haas, A.; Molnar, D., Metrical Diophantine approximation for continued fraction like maps of the interval, Trans. Am. Math. Soc., 356, 2851-2870, (2004) · Zbl 1051.11038 · doi:10.1090/S0002-9947-03-03371-3 |
| [15] | Haas, A.; Molnar, D., The distribution of jager pairs for continued fraction like mappings of the interval, Pac. J. Math, 217, 101-114, (2004) · Zbl 1080.11060 · doi:10.2140/pjm.2004.217.101 |
| [16] | Hensley, D., Continued fraction Cantor sets, Hausdorff dimension, and functional analysis, J. Number Theory, 40, 336-358, (1992) · Zbl 0745.28005 · doi:10.1016/0022-314X(92)90006-B |
| [17] | Hensley, D., Metric Diophantine approximation and probability, New York J. Math., 4, 249-257, (1998) · Zbl 0991.11046 |
| [18] | Ionescu Tulcea, C. T.; Marinescu, G., Théorie ergodique pour des classes d’opérations non complètement continues, Ann. Math., Ser. 2, 52, 140-147, (1950) · Zbl 0040.06502 · doi:10.2307/1969514 |
| [19] | S. Lang, Introduction to Diophantine Approximations (Addison-Wesley, Reading, MA, 1966), Addison-Wesley Ser. Math. · Zbl 0144.04005 |
| [20] | D. H. Mayer, The Ruelle-Araki Transfer Operator in Classical Statistical Mechanics (Springer, Berlin, 1980), Lect. Notes Phys. 123. · Zbl 0454.60079 |
| [21] | Mayer, D.; Roepstorff, G., On the relaxation time of gauss’s continued-fraction map. I: the Hilbert space approach (koopmanism), J. Stat. Phys., 47, 149-171, (1987) · Zbl 0658.10057 · doi:10.1007/BF01009039 |
| [22] | Mayer, D.; Roepstorff, G., On the relaxation time of gauss’ continued-fraction map. II: the Banach space approach (transfer operator method), J. Stat. Phys., 50, 331-344, (1988) · Zbl 0658.10058 · doi:10.1007/BF01022997 |
| [23] | Nair, R., On polynomials in primes and J. bourgain’s circle method approach to ergodic theorems. II, Stud. Math., 105, 207-233, (1993) · Zbl 0871.11051 · doi:10.4064/sm-105-3-207-233 |
| [24] | Nair, R., On the metrical theory of continued fractions, Proc. Am. Math. Soc., 120, 1041-1046, (1994) · Zbl 0796.11031 · doi:10.1090/S0002-9939-1994-1176073-5 |
| [25] | Nair, R., On uniformly distributed sequences of integers and Poincaré recurrence. II, Indag. Math., New Ser., 9, 405-415, (1998) · Zbl 0922.11065 · doi:10.1016/S0019-3577(98)80008-6 |
| [26] | Nair, R., On metric Diophantine approximation and subsequence ergodic theory, New York J. Math, 3A, 117-124, (1998) · Zbl 0894.11032 |
| [27] | Nair, R.; Weber, M., On random perturbation of some intersective sets, Indag. Math., New Ser., 15, 373-381, (2004) · Zbl 1065.37004 · doi:10.1016/S0019-3577(04)80006-5 |
| [28] | Nakada, H., Metrical theory for a class of continued fraction transformations and their natural extensions, Tokyo J. Math., 4, 399-426, (1981) · Zbl 0479.10029 · doi:10.3836/tjm/1270215165 |
| [29] | Rudolfer, S. M.; Wilkinson, K. M., A number-theoretic class of weak Bernoulli transformations, Math. Syst. Theory, 7, 14-24, (1973) · Zbl 0258.10031 · doi:10.1007/BF01824802 |
| [30] | Ruelle, D., Dynamical zeta functions and transfer operators, Notices Am. Math. Soc., 49, 887-895, (2002) · Zbl 1126.37305 |
| [31] | D. Ruelle, Thermodynamic Formalism: The Mathematical Structures of Equilibrium Statistical Mechanics, 2nd ed. (Cambridge Univ. Press, Cambridge, 2004). · Zbl 1062.82001 · doi:10.1017/CBO9780511617546 |
| [32] | A. Tempelman, Ergodic Theorems for Group Actions: Informational and Thermodynamical Aspects (Kluwer, Dordrecht, 1992), Math. Appl. 78. · Zbl 0753.28014 · doi:10.1007/978-94-017-1460-0 |
| [33] | L. Vepštas, “The Gauss-Kuzmin-Wirsing operator,” http://67.198.37.16/math/gkw.pdf |
| [34] | P. Walters, An Introduction to Ergodic Theory (Springer, New York, 1981). · Zbl 0475.28009 |
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