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On the fractional dimension of sets of continued fractions

  • Tomasz Łuczak (a1)

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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

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1.Besicovitch, A. S.. On rational approximation to real numbers. J. London Math. Soc., 9 (1934), 126131.
2.Cusick, T. W.. Hausdorff dimension of sets of continued fractions. Quart. J. Math. Oxford, 41 (1990), 277286.
3.Falconer, K.. Fractal Geometry (Wiley, Chichester, 1990).
4.Good, I. J.. The fractional dimension theory of continued fractions. Proc. Camb. Phil. Soc., 37 (1941), 199228.
5.Hirst, K. E.. A problem in the fractional dimension theory of continued fractions. Quart. J. Math. Oxford, 21 (1970), 2935.
6.Jarnik, V.. Zur metrischen Theorie der Diophantischen Approximationen. Prace Mat.-Fiz., 36 (1928), 91106.
7.Jarnik, V.. Diophantische Approximation und Hausdorffsches Mass. Rec. Math. Soc. Math. Moscou, 36 (1929), 371382.
8.Moorthy, C. G.. A problem of Good on Hausdorff dimension. Mathematiku, 39 (1992), 244246.
9.Rogers, C. A.. Some sets of continued fractions. Proc. London Math. Soc., 14 (1964), 2944.
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On the fractional dimension of sets of continued fractions

  • Tomasz Łuczak (a1)

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