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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On a problem of W. J. LeVeque concerning metric diophantine approximation

Author: Michael Fuchs
Journal: Trans. Amer. Math. Soc. 355 (2003), 1787-1801
MSC (2000): Primary 11J83, 60F05
Published electronically: December 18, 2002
MathSciNet review: 1953525
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Abstract: We consider the diophantine approximation problem

\begin{displaymath}\left\vert x-\frac{p}{q}\right\vert\leq\frac{f(\log q)}{q^2} \end{displaymath}

where $f$ is a fixed function satisfying suitable assumptions. Suppose that $x$ is randomly chosen in the unit interval. In a series of papers that appeared in earlier issues of this journal, LeVeque raised the question of whether or not the central limit theorem holds for the solution set of the above inequality (compare also with some work of Erdos). Here, we are going to extend and solve LeVeque's problem.

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Additional Information

Michael Fuchs
Affiliation: Institut für Geometrie, Technische Universität Wien, Wiedner Hauptstrasse 8-10/113, 1040 Wien, Austria
Address at time of publication: Institute of Statistical Science, Academia Sinica, Taipei, 115, Taiwan, R.O.C.

Keywords: Continued fractions, metric diophantine approximation, dependent random variables, central limit theorem
Received by editor(s): February 7, 2002
Received by editor(s) in revised form: September 18, 2002
Published electronically: December 18, 2002
Additional Notes: This work was supported by the Austrian Science Foundation FWF, grant S8302-MAT
Article copyright: © Copyright 2002 American Mathematical Society