On the fractional dimension of sets of continued fractions
On the fractional dimension of sets of continued fractions
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DOI：
10.1112/s0025579300011955
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发表时间：
1997-06
期刊：
影响因子：
0.8
通讯作者：
Tomasz Łuczak
中科院分区：
文献类型：
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作者：
Tomasz Łuczak
让[0;A 1 (ξ), A 2 (ξ)，…]表示ξ∈[0,1]的连分式展开式。估计连分数集的分数维数的问题出现在20年代末Jarnik[6,7]和Besicovitch[1]的论文中，从那时起，许多作者已经解决了这个问题（参见[2,4,5,8,9]）。特别地，Good[4]证明了所有ξ的集合，其中n （ξ）→∞为n→∞时，其Hausdorff维数为1 / 2。对于展开项为双指数无穷大的连分式集合，其维数进一步减小。更准确地说，让赫斯特[5]表示暗淡，另一方面，穆尔西[8]表示暗淡的地方
Let [0; a 1 (ξ), a 2 (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 a n (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