Lower bound on the Hausdorff dimension of a set of complex continued fractions

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• 作者：Amit Priyadarshi ^{priyadarshi@maths.iitd.ac.in}
• 关键词：Hausdorff dimension ; Continued fractions ; Iterated function systems ; Perron&ndash ; Frobenius operators ; Spectral radius
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 May 2017
• 年：2017
• 卷：449
• 期：1
• 页码：91-95
• 全文大小：250 K
• 卷排序：449

文摘

Let J   be the set of all infinite complex continued fractions with partial numerators equal to one and partial denominators b1,b2,…b1,b2,…, where each bkbk is a complex number of the form m+nim+ni with m being a positive integer and n being any integer. In this paper we give an algorithm to determine lower bounds for the Hausdorff dimension of J. We show that the Hausdorff dimension of J is greater than 1.825, which is the best known lower bound, to the best of our knowledge. The ideas used to obtain this improved bound is different from those used by other authors and could be applied to estimate Hausdorff dimensions of other sets as well.