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

DOI:
https://doi.org/10.1090/S0002-9947-99-02272-2

Published electronically:
June 10, 1999

MathSciNet review:
1491854

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Abstract | References | Similar Articles | Additional Information

Abstract: For let be a Cantor set constructed from the interval , and let . We derive conditions under which

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 , where is a set of positive integers and is the set of real numbers such that all partial quotients of , except possibly the first, are members of .

**1.**S. Astels.*Cantor sets and numbers with restricted partial quotients*(Ph.D. thesis), University of Waterloo, 1999.**2.**T. W. Cusick.*Sums and products of continued fractions*, Proc. Amer. Math. Soc., 27 (1971), 35-38. MR**42:4498****3.**T. W. Cusick and R. A. Lee.*Sums of sets of continued fractions*, Proc. Amer. Math. Soc., 30 (1971), 241-246. MR**44:158****4.**Bohuslav Divi\v{s}.*On the sums of continued fractions*, Acta Arith., 22 (1973), 157-173. MR**51:8043****5.**Marshall Hall, Jr.*On the sum and product of continued fractions*, Ann. of Math., 48 (1947), 966-993. MR**9:226b****6.**James L. Hlavka.*Results on sums of continued fractions*, Trans. Amer. Math. Soc., 211 (1975), 123-134. MR**51:12720****7.**Peter R. Massopust.*Fractal functions, fractal surfaces, and wavelets*, Academic Press, 1994. MR**96b:28007****8.**Sheldon E. Newhouse.*The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms*, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 101-151. MR**82e:58067****9.**J. Palis and F. Takens.*Hyperbolicity & sensitive chaotic dynamics at homoclinic bifurcations*, Cambridge University Press, 1993. MR**94h:58129**

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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:
https://doi.org/10.1090/S0002-9947-99-02272-2

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