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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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: We initiate the study of the sets $ \mathcal{H}(c)$, $ 0<c<1$, of real $ x$ for which the sequence $ (kx)_{k\geq1}$ (viewed mod 1) consistently hits the interval $ [0,c)$ at least as often as expected (i. e., with frequency $ \geq c$). More formally,

$\displaystyle \mathcal{H}(c) \stackrel{\text{def}}{=} \big\{\alpha\in \mathbb{... ...n\mid \langle k\alpha\rangle <c\}\big)\geq cn, \text{ for all } n\geq1\big\}, $

where $ \langle x\rangle =x-[x]$ stands for the fractional part of $ x\in\mathbb{R}$.

We prove that, for rational $ c$, the sets $ \mathcal{H}(c)$ are of positive Hausdorff dimension and, in particular, are uncountable. For integers $ m\geq1$, we obtain a surprising characterization of the numbers $ \alpha\in\mathcal{H}_m \stackrel{\text{def}}{=} \mathcal{H}(\tfrac1m)$ in terms of their continued fraction expansions: The odd entries (partial quotients) of these expansions are divisible by $ m$. The characterization implies that $ x\in\mathcal{H}_m$ if and only if $ \frac 1{mx} \in\mathcal{H}_m$, for $ x>0$. We are unaware of a direct proof of this equivalence without making use of the mentioned characterization of the sets $ \mathcal{H}_m$.

We also introduce the dual sets $ \widehat{\mathcal{H}}_m$ of reals $ y$ for which the sequence of integers $ \big([ky]\big)_{k\geq1}$ consistently hits the set $ m\mathbb{Z}$ with the at least expected frequency $ \frac1m$ and establish the connection with the sets $ \mathcal{H}_m$:

\begin{displaymath}\text{ If $xy=m$ for $x,y>0$, then $x\in\mathcal{H}_m\iff y\in\widehat{\mathcal{H}}_m$.} \end{displaymath}

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


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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
PII: S 0002-9939(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.