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

Author(s): Michael Fuchs
Journal: Trans. Amer. Math. Soc. 355 (2003), 1787-1801.
MSC (2000): Primary 11J83, 60F05
Posted: December 18, 2002
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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.
Email: fuchs@stat.sinica.edu.tw

DOI: 10.1090/S0002-9947-02-03225-7
PII: S 0002-9947(02)03225-7
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
Posted: December 18, 2002
Additional Notes: This work was supported by the Austrian Science Foundation FWF, grant S8302-MAT
Copyright of article: Copyright 2002, American Mathematical Society

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