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Zbl 0689.10060
Hensley, Doug
The Hausdorff dimensions of some continued fraction Cantor sets.
(English)
[J] J. Number Theory 33, No.2, 182-198 (1989). ISSN 0022-314X; ISSN 1096-1658/e

The paper deals with the estimation of the Hausdorff dimension of sets $E\sb n=E(x \vert a\sb 1,a\sb 2,...\le n)$, where $x=(0;a\sb 1a\sb 2...)$ is the continued fraction expansion of x. According to a result of Th. Cusick, the dimension is characterized by the convergence-exponent of the series $$(1)\quad \sum\sp{\infty}\sb{r=0}\sum\sb{\nu \in B\sb n(r)}\frac{1}{\nu\sp s}\quad,$$ where the inner sum is carried out over the B's with $(0;a\sb 1a\sb 2...a\sb r)=A\sb r/B\sb r$ with $(A\sb r,B\sb r)=1$ and $a\sb 1,a\sb 2,...,a\sb r\le n.$ \par The novelty in the paper is the use of a recursion-formula making the use of computers more efficient in the calculating of the convergence- exponent of the series (1).
[P.Szüsz]
MSC 2000:
*11K55 Metric theory of other number-theoretic algorithms and expansions
28D99 Measure-theoretic ergodic theory
11J70 Continued fractions and generalizations

Keywords: Cantor sets; Hausdorff dimension; continued fraction expansion; convergence-exponent

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