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

Full-text PDF Free Access

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**0269603**, 10.1090/S0002-9939-1971-0269603-3**3.**T. W. Cusick and R. A. Lee,*Sums of sets of continued fractions*, Proc. Amer. Math. Soc.**30**(1971), 241–246. MR**0282924**, 10.1090/S0002-9939-1971-0282924-3**4.**Bohuslav Diviš,*On the sums of continued fractions*, Acta Arith.**22**(1973), 157–173. MR**0371826****5.**Marshall Hall Jr.,*On the sum and product of continued fractions*, Ann. of Math. (2)**48**(1947), 966–993. MR**0022568****6.**James L. Hlavka,*Results on sums of continued fractions*, Trans. Amer. Math. Soc.**211**(1975), 123–134. MR**0376545**, 10.1090/S0002-9947-1975-0376545-X**7.**Peter R. Massopust,*Fractal functions, fractal surfaces, and wavelets*, Academic Press, Inc., San Diego, CA, 1994. MR**1313502****8.**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****9.**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**

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:
http://dx.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