ON THE DENSITY OF HAUSDORFF DIMENSIONS OF BOUNDED TYPE CONTINUED FRACTION SETS: THE TEXAN CONJECTURE
ON THE DENSITY OF HAUSDORFF DIMENSIONS OF BOUNDED TYPE CONTINUED FRACTION SETS: THE TEXAN CONJECTURE
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DOI：
10.1142/s0219493704000900
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发表时间：
2004-03
影响因子：
1.1
通讯作者：
O. Jenkinson
中科院分区：
文献类型：
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作者：
O. Jenkinson
给定自然数的一个非空有限子集A，设EA表示其连分数位在A中的无理数x∈[0，1]的集合.一般而言，EA是其Hausdorff维数dim(EA)介于0和1之间的康托集.然后，我们描述了一种精确计算尺寸DIM(Ea)的方法，并用它数值研究了[1/2，1]相交的方式。这些计算倾向于支持由Hensley和Mauldin&Urbanski独立提出的猜想，该猜想在[0，1]中是稠密的。在重要的特例A={1，2}中，我们用我们的计算方法给出了DIM(E{1，2})的精确近似，改进了文[18]中给出的结果。
Given a non-empty finite subset A of the natural numbers, let EA denote the set of irrationals x∈[0,1] whose continued fraction digits lie in A. In general, EA is a Cantor set whose Hausdorff dimension dim(EA) is between 0 and 1. It is shown that the set intersects [0,1/2] densely. We then describe a method for accurately computing dimensions dim(EA), and employ it to investigate numerically the way in which intersects [1/2,1]. These computations tend to support the conjecture, first formulated independently by Hensley, and by Mauldin & Urbanski, that is dense in [0,1]. In the important special case A={1,2}, we use our computational method to give an accurate approximation of dim(E{1,2}), improving on the one given in [18].