Continued fractions and heavy sequences

Authors:
Michael Boshernitzan and David Ralston

Journal:
Proc. Amer. Math. Soc. **137** (2009), 3177-3185

MSC (2000):
Primary 11K38, 11J71, 37A30

Published electronically:
May 15, 2009

MathSciNet review:
2515388

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Abstract | References | Similar Articles | Additional Information

Abstract: We initiate the study of the sets , , of real for which the sequence (viewed mod 1) consistently hits the interval at least as often as expected (i. e., with frequency ). More formally,

We prove that, for rational , the sets are of positive Hausdorff dimension and, in particular, are uncountable. For integers , we obtain a surprising characterization of the numbers in terms of their continued fraction expansions: The odd entries (partial quotients) of these expansions are divisible by . The characterization implies that if and only if , for . We are unaware of a direct proof of this equivalence without making use of the mentioned characterization of the sets .

We also introduce the dual sets of reals for which the sequence of integers consistently hits the set with the at least expected frequency and establish the connection with the sets :

The motivation for the present study comes from Y. Peres's ergodic lemma.

**1.**Y. Dupain and Vera T. Sós,*On the one-sided boundedness of discrepancy-function of the sequence {𝑛𝛼}*, Acta Arith.**37**(1980), 363–374. MR**598889****2.**Kenneth Falconer,*Fractal geometry*, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR**1102677****3.**H. Furstenberg,*Recurrence in ergodic theory and combinatorial number theory*, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR**603625****4.**A. Ya. Khinchin,*Continued fractions*, The University of Chicago Press, Chicago, Ill.-London, 1964. MR**0161833****5.**Serge Lang,*Introduction to Diophantine approximations*, 2nd ed., Springer-Verlag, New York, 1995. MR**1348400****6.**Yuval Peres,*A combinatorial application of the maximal ergodic theorem*, Bull. London Math. Soc.**20**(1988), no. 3, 248–252. MR**931186**, 10.1112/blms/20.3.248**7.**D. Ralston, Heaviness--An Extension of a Lemma of Y. Peres,*Houston Journal of Mathematics*. To appear.

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Additional Information

**Michael Boshernitzan**

Affiliation:
Department of Mathematics, Rice University, Houston, Texas 77005

Email:
michael@rice.edu

**David Ralston**

Affiliation:
Department of Mathematics, Rice University, Houston, Texas 77005

Address at time of publication:
Department of Mathematics, Ohio State University, 231 W. 18th Avenue, Columbus, Ohio 43210

DOI:
http://dx.doi.org/10.1090/S0002-9939-09-09625-7

Received by editor(s):
October 24, 2007

Received by editor(s) in revised form:
March 7, 2008

Published electronically:
May 15, 2009

Additional Notes:
The second author was supported in part by NSF VIGRE grant DMS–0240058.

Communicated by:
Michael T. Lacey

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.