Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-28T17:20:43.492Z Has data issue: false hasContentIssue false
  • English
  • Français

On the metric theory of the optimal continued fraction expansion

Published online by Cambridge University Press:  17 April 2009

R. Nair
R. Nair
Affiliation:
Department of Pure MathematicsUniversity of LiverpoolPO Box 147Liverpool L69 3BXUnited Kingdom
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Suppose kn denotes either φ(n) or φ(rn) (n = 1, 2, …) where the polynomial φ maps the natural numbers to themselves and rk denotes the kth rational prime. Let denote the sequence of convergents to a real numbers x for the optimal continued fraction expansion. Define the sequence of approximation constants by

In this paper we study the behaviour of the sequence for all most all x with respect to Lebesgue measure. In the special case where kn = n (n = 1, 2, …) these results are due to Bosma and Kraaikamp.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 56 , Issue 1 , August 1997 , pp. 69 - 79
Copyright
Copyright © Australian Mathematical Society 1997

References

[1]Bosma, W., ‘Optimal continued fractions’, Indag. Math. 49 (1987), 353–379.Google Scholar
[2]Bosma, W. and Kraaikamp, C., ‘Metrical theory for optimal continued fractions’, J. Number Theory 34 (1990), 251–270.Google Scholar
[3]Bourgain, J., ‘On the maximal ergodic theorem for certain subsets of the integers’, Israel. J. Math. 61 (1988), 39–72.Google Scholar
[4]Bourgain, J., ‘Pointwise ergodic theorems for arithmetic sets’, Inst. Hautes Études Sci. Publ. Math. 69 (1989), 5–45.Google Scholar
[5]Cornfeld, I.P., Formin, S.V. and Sinai, Ya. G., Ergodic theory (Springer-Verlag, Berlin, Heidelberg, New York, 1982).CrossRefGoogle Scholar
[6]Hardy, G.H. and Wright, E., An introduction to number theory (Oxford University Press, New York, 1979).Google Scholar
[7]Ito, S., Nakada, H. and Tanaka, S., ‘On the invariant measure for the transformation associated with some real continued fractions’, Keio Engineering Reports 30 (1981), 61–69.Google Scholar
[8]Kraaikramp, C., ‘Maximal S-expansions are Bernoulli shifts’, Bull. Soc. Math. France (to appear).Google Scholar
[9]Minnigerode, B., ‘Über eine neue Methode die Pellsche Gleichung auf zielösen’, Nachr. Göttingen (1873).Google Scholar
[10]Minkowski, H., ‘Über die Annäherung an eine reelle Grösse durch rationale Zahlen’, Math. Ann. 54 (1901), 91–124.Google Scholar
[11]Nair, R., ‘On the metrical theory of continued fractions’, Proc. Amer. Math. Soc. 120 (1994).CrossRefGoogle Scholar
[12]Nair, R., ‘On Polynomials in primes and J. Bourgains's circle method approach to ergodic theorems II’, Studia Math. 105 (1993), 207–233.Google Scholar
[13]Perron, O., Die Lehre von der Kettenbrüche, Band 1, 3Auft (B–G Teubner, Stuttgart, 1954).Google Scholar
[14]Rieger, G.J., ‘Mischung und Ergodizität bei Kettenbrüche nach nächsten Ganzen’, J. Reine Angew. Math. 310 (1979), 171–181.Google Scholar
[15]Rieger, G.J., ‘Über die Länge von Kettenbtrüchen mit ungeraden Teilnennern’, Abh. Braunschweig. Wiss. Ges. 32 (1981), 61–69.Google Scholar
[16]Rhin, G., ‘Sur la répartition modulo 1 de suites f(p)’ Acta. Arith. 23 (1973), 217–248.Google Scholar
[17]Walters, P., An introduction to ergodic theory, Graduate Texts in Mathematics 79 (Springer Verlag, Berlin, Heidelberg, New York, 1982).Google Scholar
[18]Weyl, H., ‘Über die Gleichvertilung von Zahlen mod. Eins’, Math. Ann. 77 (1916), 313–361.Google Scholar