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On the fractional dimension of sets of continued fractions

Published online by Cambridge University Press:  26 February 2010

Tomasz Łuczak
Affiliation:
Department of Mathematics, Adam Mickiewicz University, Poznań, Poland.

Extract

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

Type
Research Article
Copyright
Copyright © University College London 1997

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References

1.Besicovitch, A. S.. On rational approximation to real numbers. J. London Math. Soc., 9 (1934), 126131.CrossRefGoogle Scholar
2.Cusick, T. W.. Hausdorff dimension of sets of continued fractions. Quart. J. Math. Oxford, 41 (1990), 277286.CrossRefGoogle Scholar
3.Falconer, K.. Fractal Geometry (Wiley, Chichester, 1990).Google Scholar
4.Good, I. J.. The fractional dimension theory of continued fractions. Proc. Camb. Phil. Soc., 37 (1941), 199228.CrossRefGoogle Scholar
5.Hirst, K. E.. A problem in the fractional dimension theory of continued fractions. Quart. J. Math. Oxford, 21 (1970), 2935.CrossRefGoogle Scholar
6.Jarnik, V.. Zur metrischen Theorie der Diophantischen Approximationen. Prace Mat.-Fiz., 36 (1928), 91106.Google Scholar
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8.Moorthy, C. G.. A problem of Good on Hausdorff dimension. Mathematiku, 39 (1992), 244246.CrossRefGoogle Scholar
9.Rogers, C. A.. Some sets of continued fractions. Proc. London Math. Soc., 14 (1964), 2944.CrossRefGoogle Scholar

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