Tytuł artykułu
Autorzy
Wybrane pełne teksty z tego czasopisma
Identyfikatory
Warianty tytułu
Języki publikacji
Abstrakty
We find upper and lower estimates on the Hausdorff dimension of the set of real numbers which have coeffcients in a generalized continued fraction expansion that are bounded by a constant. As a consequence we prove a version of Jarník's theorem: the set of real numbers with bounded coeffcients in their generalized continued fraction representation has Hausdorff dimension one.
Wydawca
Rocznik
Tom
Strony
115--122
Opis fizyczny
Bibliogr. 16 poz., tab.
Twórcy
autor
- Leibniz Universität Hannover, 30167 Hannover, Germany
Bibliografia
- [1] J. M. Borwein, K. G. Hare and J. G. Lynch, Generalized continued logarithms and related continued fractions, arXiv:1606.06984 (2016).
- [2] R. T. Bumby, Hausdorff dimension of Cantor sets, J. Reine Angew. Math. 331 (1982), 192-206.
- [3] R. T. Bumby, Hausdorff Dimension of Sets Arising in Number Theory, Lecture Notes in Math. 1135, Springer, New York, 1985.
- [4] H.-C. Chan, The asymptotic growth rate of random Fibonacci type sequences I, Fibonacci Quart. 43 (2005), 243-255.
- [5] H.-C. Chan, The asymptotic growth rate of random Fibonacci type sequences II, Fibonacci Quart. 44 (2006), 73-84.
- [6] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, New York, 1990.
- [7] I. J. Good, The fractional dimension of continued fractions, Proc. Cambridge Philos. Soc. 37 (1941), 199-208.
- [8] D. Hensley, The Hausdorff dimension of some continued fraction Cantor sets, J. Number Theory 33 (1989), 182-198.
- [9] D. Hensley, A polynomial time algorithm for the Hausdorff dimension of continued fraction Cantor sets, J. Number Theory 58 (1996), 9-45.
- [10] V. Jarník, Zur metrischen Theorie der diophantischen Approximationen, Prace Matematyczno-Fizyczne 36 (1929), 91-106.
- [11] O. Jenkinson and M. Pollicott, Computing the dimension of dynamically defined sets: E2 and bounded continued fractions, Ergodic Theory Dynam. Systems 21 (2001), 1429-1445.
- [12] A. Khintchine, Zur metrischen Kettenbruchtheorie, Compos. Math. 3 (1936), 275-285.
- [13] D. Lascu, A Gauss-Kuzmin-type problem for a family of continued fraction expansions, J. Number Theory 133 (2013), 2153-2181.
- [14] D. Lascu, A Gauss-Kuzmin theorem for continued fractions associated with nonpositive integer powers of an integer b ≥ 2, Scientific World J. 2014, 8 pp.
- [15] R. D. Mauldin and M. Urbański, Dimensions and measures in iterated function systems, Proc. London Math. Soc. (3) 73 (1996), 105-154.
- [16] Ya. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications, Univ. of Chicago Press, Chicago, 1997.
Uwagi
Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-1144c516-001a-42f5-853d-6dc8fde136b2