Thickness measures for Cantor sets
Author:
S. Astels
Journal:
Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 108-111
MSC (1991):
Primary 58F12; Secondary 28A78
DOI:
https://doi.org/10.1090/S1079-6762-99-00068-2
Published electronically:
July 20, 1999
MathSciNet review:
1701889
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Abstract: For a fixed $k\ge 1$ let $C_1,\dots ,C_k$ be generalized Cantor sets. We examine various criteria under which $C_1+\dots + C_k$ contains an interval. When these criteria do not hold, we give a lower bound for the Hausdorff dimension of $C_1+\dots +C_k$. Our work will involve the development of two different types of thickness measures.
- S. Astels. Cantor sets and numbers with restricted partial quotients, Trans. Amer. Math. Soc. (to appear).
- Thomas W. Cusick and Mary E. Flahive, The Markoff and Lagrange spectra, Mathematical Surveys and Monographs, vol. 30, American Mathematical Society, Providence, RI, 1989. MR 1010419
- Sheldon E. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 101–151. MR 556584
- Jacob Palis and Floris Takens, Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Cambridge Studies in Advanced Mathematics, vol. 35, Cambridge University Press, Cambridge, 1993. Fractal dimensions and infinitely many attractors. MR 1237641
- S. Astels. Cantor sets and numbers with restricted partial quotients, Trans. Amer. Math. Soc. (to appear).
- Thomas W. Cusick and Mary E. Flahive. The Markoff and Lagrange spectra, Amer. Math. Soc., Providence, RI, 1989.
- 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.
- J. Palis and F. Takens. Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Cambridge University Press, Cambridge, 1993.
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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
Keywords:
Cantor sets,
sums of sets,
Hausdorff dimension
Received by editor(s):
March 15, 1999
Published electronically:
July 20, 1999
Additional Notes:
Research supported in part by the Natural Sciences and Engineering Research Council of Canada.
Communicated by:
Yitzhak Katznelson
Article copyright:
© Copyright 1999
American Mathematical Society
