Abstract
We use the moving average ergodic theorem of A. Bellow, R. Jones and J. Rosenblatt to derive various results in metric number theory primarily concerning moving averages of various sequences attached to the optimal continued fraction expansion of a real number.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
W. Bosma, Optimal continued fractions, Indag. Math., 49 (1987), 353–379.
A. Bellow, R. Jones and J. Rosenblatt, Convergence of moving averages, Ergodic Theory Dynam. Systems, 10 (1990), 43–62.
W. Bosma, H. Jager and F. Wiedijk, Some metrical observations on the approximation by continued fractions, Indag. Math., 45 (1983), 281–299.
W. Bosma and C. Kraaikamp, Metrical theory for optimal continued fractions, J. Number Theory, 34 (1990), 251–270.
I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer Verlag, 1982.
G. H. Hardy and E. Wright, An Introduction to Number Theory, Oxford University Press, 1979.
C. Kraaikamp, Maximal S-expansions are Bernoulli shifts, Bull. Soc. Math. France, 121 (1993), 117–131.
B. Minnigerode, Über eine neue Methode die Pellsche Gleichung auf zielösen, Nachr. Göttingen (1873).
H. Minkowski, Über die Annäherung an eine reelle Grösse durch rationale Zahlen, Math. Ann., 54 (1901), 91–124.
H. Kamarul-haili and R. Nair, On moving averages and continued fractions, Uniform Distribution Theory, to appear.
O. Perron, Die Lehre von der Kettenbrüche, Band 1, Verlag von B. G. Teubner, Berlin, Leipzig, 1913.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by István Berkes
Rights and permissions
About this article
Cite this article
Haili, H.K., Nair, R. Optimal continued fractions and the moving average ergodic theorem. Period Math Hung 66, 95–103 (2013). https://doi.org/10.1007/s10998-012-7874-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10998-012-7874-5