Continued fractions and heavy sequences
Authors:
Michael Boshernitzan and David Ralston
Journal:
Proc. Amer. Math. Soc. 137 (2009), 31773185
MSC (2000):
Primary 11K38, 11J71, 37A30
Published electronically:
May 15, 2009
MathSciNet review:
2515388
Fulltext PDF Free Access
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, where stands for the fractional part of . 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 onesided boundedness of discrepancyfunction
of the sequence {𝑛𝛼}, Acta Arith. 37
(1980), 363–374. MR 598889
(82c:10058)
 2.
Kenneth
Falconer, Fractal geometry, John Wiley & Sons, Ltd.,
Chichester, 1990. Mathematical foundations and applications. MR 1102677
(92j:28008)
 3.
H.
Furstenberg, Recurrence in ergodic theory and combinatorial number
theory, Princeton University Press, Princeton, N.J., 1981. M. B.
Porter Lectures. MR 603625
(82j:28010)
 4.
A.
Ya. Khinchin, Continued fractions, The University of Chicago
Press, Chicago, Ill.London, 1964. MR 0161833
(28 #5037)
 5.
Serge
Lang, Introduction to Diophantine approximations, 2nd ed.,
SpringerVerlag, New York, 1995. MR 1348400
(96h:11067)
 6.
Yuval
Peres, A combinatorial application of the maximal ergodic
theorem, Bull. London Math. Soc. 20 (1988),
no. 3, 248–252. MR 931186
(89e:28033), http://dx.doi.org/10.1112/blms/20.3.248
 7.
D. Ralston, HeavinessAn Extension of a Lemma of Y. Peres, Houston Journal of Mathematics. To appear.
 1.
 Y. Dupain, T. Vera Sós, On the onesided boundedness of discrepancyfunction of the sequence , Acta Arith. 37 (1980), 363374. MR 598889 (82c:10058)
 2.
 K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Ltd., 1990. MR 1102677 (92j:28008)
 3.
 H. Furstenberg, Recurrence in ergodic theorem and combinatorial number theory, Princeton University Press, 1981. MR 603625 (82j:28010)
 4.
 A. Y. Khinchin, Continued Fractions, The University of Chicago Press, 1964. MR 0161833 (28:5037)
 5.
 S. Lang, Introduction to Diophantine Approximations, SpringerVerlag, 1995. MR 1348400 (96h:11067)
 6.
 Y. Peres, A combinatorial application of the maximal ergodic theorem, Bull. London Math. Soc. 20 (1988), 248252. MR 931186 (89e:28033)
 7.
 D. Ralston, HeavinessAn Extension of a Lemma of Y. Peres, Houston Journal of Mathematics. To appear.
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
11K38,
11J71,
37A30
Retrieve articles in all journals
with MSC (2000):
11K38,
11J71,
37A30
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/S0002993909096257
PII:
S 00029939(09)096257
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.
