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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 n =E(x|a 1 ,a 2 ,···n), where x=(0;a 1 a 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) r=0 νB n (r) 1 ν s ,

where the inner sum is carried out over the B’s with (0;a 1 a 2 ···a r )=A r /B r with (A r ,B r )=1 and a 1 ,a 2 ,···,a r 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