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The Hausdorff dimensions of some continued fraction Cantor sets. (English) Zbl 0689.10060
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 $a\sb 1,a\sb 2,...,a\sb r\le n.$ 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).
Reviewer: P.Szüsz

MSC:
11K55Metric theory of other number-theoretic algorithms and expansions
28D99Measure-theoretic ergodic theory
11J70Continued fractions and generalizations
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References:
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