Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the metrical theory of continued fractions


Author: R. Nair
Journal: Proc. Amer. Math. Soc. 120 (1994), 1041-1046
MSC: Primary 11K50; Secondary 28D05
DOI: https://doi.org/10.1090/S0002-9939-1994-1176073-5
MathSciNet review: 1176073
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose $ {b_k}$ denotes either $ \phi (k)$ or $ \phi ({p_k})\;(k = 1,2, \ldots )$ where the polynomial $ \phi $ maps $ \mathbb{N}$ to $ \mathbb{N}$ and $ {p_k}$ denotes the $ k$th rational prime. Suppose $ ({c_k}(x))_{k = 1}^\infty $ denotes the sequences of partial quotients of the continued function expansion of the real number $ x$. Then for certain functions $ F:{\mathbb{R}_{ \geqslant 0}} \to \mathbb{R}$ we show that

$\displaystyle \mathop {\lim }\limits_{N \to \infty } {F^{ - 1}}\left[ {\frac{{F... ...ft[ {\frac{1} {{(\log 2)}}\int_0^1 {\frac{{F({c_1}(x))}} {{1 + x}}dx} } \right]$

almost everywhere with respect to Lebesgue measure. This result with $ {b_k} = k$ is classical and due to Ryll-Nardzewski.

References [Enhancements On Off] (What's this?)

  • [1] A. Bellow and V. Losert, The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences, Trans. Amer. Math. Soc. 288 (1985), 307-345.
  • [2] G. A. Edgar and L. Sucheston (eds.), Almost everywhere convergence, Academic Press, New York, 1989, pp. 145-151.
  • [3] Y. Katznelson, An introduction to harmonic analysis, Dover, New York, 1976.
  • [4] A. Khinchine, Metrische Kettenbruch probleme, Compositio Math. 1 (1935), 359-382.
  • [5] P. Lévi, Sur le dévelopment en fraction continue d'un nombre choise au hasard, Compositio Math. 3 (1936), 286-303.
  • [6] R. Nair, On polynomials in primes and J. Bourgain's circle method approach to ergodic theorems. II, Studia Math. (to appear).
  • [7] K. Peterson, Ergodic theory, Graduate Stud. in Adv. Math., vol. 2, Cambridge Univ. Press, London and New York, 1981.
  • [8] G. Rhin, Répartition modulo $ 1$ de $ f({p_n})$ quand $ f$ est une series entiére, Lecture Notes in Math., vol. 475, Springer-Verlag, New York, 1975, pp. 176-244. MR 0392857 (52:13670)
  • [9] C. Ryll-Nardzewski, On the ergodic theorems. II: Ergodic theory of continued fractions, Studia Math. 12 (1951), 74-79. MR 0046583 (13:757b)
  • [10] P. Walters, An introduction to ergodic theory, Graduate Texts in Math., vol. 79, Springer-Verlag, New York, 1982. MR 648108 (84e:28017)
  • [11] H. Weyl, Über die Gleichvertilung von Zahlen mod. Eins, Math. Ann. 77 (1916), 313-352. MR 1511862

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11K50, 28D05

Retrieve articles in all journals with MSC: 11K50, 28D05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1176073-5
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society