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On the fractional dimension of sets of continued fractions

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  26 February 2010

Tomasz Łuczak
Tomasz Łuczak
Affiliation:
Department of Mathematics, Adam Mickiewicz University, Poznań, Poland.
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Let [0;a1(ξ), a2(ξ),…] denote the continued fraction expansion of ξ∈[0, 1]. The problem of estimating the fractional dimension of sets of continued fractions emerged in late twenties in papers by Jarnik [6, 7] and Besicovitch [1] and since then has been addressed by a number of authors (see [2, 4, 5, 8, 9]). In particular, Good [4] proved that the set of all ξ, for which an(ξ)→∞ as n→∞ has the Hausdorff dimension ½ For the set of continued fractions whose expansion terms tend to infinity doubly exponentially the dimension decreases even further. More precisely, let

Hirst [5] showed that dim On the other hand, Moorthy [8] showed that dim where

MSC classification

Secondary: 11J70: Continued fractions and generalizations
Type
Research Article
Information
Mathematika , Volume 44 , Issue 1 , June 1997 , pp. 50 - 53
Copyright
Copyright © University College London 1997

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References

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