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Some metrical properties of continued fractions

Published online by Cambridge University Press:  26 February 2010

G. Ramharter
G. Ramharter
Affiliation:
Institut fuer Analysis, Techn. Universitaet Wien, A-1040 Wien, Gusshausstrasse 27–29, Austria
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§1. Let E = E(A) be the set of real numbers x ε (0, 1) whose regular continued fraction expansion

contains only partial denominators ai from a given set A of positive integers. For finite A the (Hausdorff-) dimensional numbers dim E have been studied by I. J. Good ‘2’ and T. W. Cusick ‘1’. C. A. Rogers ‘8’ introduced a natural probability measure on E. He showed that excluding sets with measure between 0 and 1 (in the strict sense) from E does not reduce the dimensionality more than excluding sets of measure zero, and that the minimal (“essential”) dimensions ess dim E arising in this way is smaller than dim E, at least for A = {1,…, r} when r = 2 or r is sufficiently large.

Type
Research Article
Information
Mathematika , Volume 30 , Issue 1 , June 1983 , pp. 117 - 132
Copyright
Copyright © University College London 1983

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References

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