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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Cantor sets and numbers
with restricted partial quotients


Author: S. Astels
Journal: Trans. Amer. Math. Soc. 352 (2000), 133-170
MSC (1991): Primary 11J70, 58F12; Secondary 11Y65, 28A78
Published electronically: June 10, 1999
MathSciNet review: 1491854
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Abstract: For $j=1,\dots,k$ let $C_j$ be a Cantor set constructed from the interval $I_j$, and let $\epsilon _j=\pm 1$. We derive conditions under which

\begin{equation*}\epsilon _1 C_1+\dots+\epsilon _k C_k = \epsilon _1 I_1+\dots+\epsilon _k I_k \quad\text{and}\quad C_1^{\epsilon _1}\dotsb C_k^{\epsilon _k}= I_1^{\epsilon _1}\dotsb I_k^{\epsilon _k}.\end{equation*}

When these conditions do not hold, we derive a lower bound for the Hausdorff dimension of the above sum and product. We use these results to make corresponding statements about the sum and product of sets $F(B_j)$, where $B_j$ is a set of positive integers and $F(B_j)$ is the set of real numbers $x$ such that all partial quotients of $x$, except possibly the first, are members of $B_j$.


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Additional Information

S. Astels
Affiliation: Department of Pure Mathematics, The University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: sastels@barrow.uwaterloo.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02272-2
PII: S 0002-9947(99)02272-2
Keywords: Continued fractions, Cantor sets, sums of sets, Hausdorff dimension
Received by editor(s): July 3, 1997
Received by editor(s) in revised form: December 15, 1997
Published electronically: June 10, 1999
Additional Notes: Research supported in part by the Natural Sciences and Engineering Research Council of Canada
Article copyright: © Copyright 1999 American Mathematical Society