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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

11K55Metric theory of other number-theoretic algorithms and expansions
28D99Measure-theoretic ergodic theory
11J70Continued fractions and generalizations
Full Text: DOI
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[2] Cusick, T. W.: Continuants with bounded digits. Mathematika 24, 166-172 (1977) · Zbl 0371.10023
[3] Cusick, T. W.: Continuants with bounded digits II. Mathematika 25, 107-109 (1978) · Zbl 0388.10021
[4] Cusick, T. W.: Continuants with bounded digits III. Mh. math. 99, 105-109 (1985) · Zbl 0574.10035
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[6] Hensley, D.: The distribution of badly approximable rationals. Proceedings, int. Number theory conference (1989) · Zbl 0689.10042
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[8] Jarnik, I.: Zur metrischen theorie der diophantischen approximationen. Proc. mat. Fyz. 36, 91-106 (1928)