Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Cantor sets and numbers with restricted partial quotients

Author(s): S. Astels
Journal: Trans. Amer. Math. Soc. 352 (2000), 133-170.
MSC (1991): Primary 11J70, 58F12; Secondary 11Y65, 28A78
Posted: June 10, 1999
MathSciNet review: 1491854
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: For $j=1,\dots,k$ let be a Cantor set constructed from the interval , 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 , 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 .

References:

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

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 11J70, 58F12, 11Y65, 28A78

Retrieve articles in all Journals with MSC (1991): 11J70, 58F12, 11Y65, 28A78

Additional Information:

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

DOI: 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
Posted: June 10, 1999
Additional Notes: Research supported in part by the Natural Sciences and Engineering Research Council of Canada
Copyright of article: Copyright 1999, American Mathematical Society

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|