Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T19:11:15.758Z Has data issue: false hasContentIssue false
  • English
  • Français

A Technique of Studying Sums of Central Cantor Sets

Published online by Cambridge University Press:  20 November 2018

Razvan Anisca  and
Monica Ilie
Razvan Anisca
Affiliation:
Department of Mathematical Sciences University of Alberta Edmonton, Alberta T6G 2G1
Monica Ilie
Affiliation:
Department of Mathematical Sciences University of Alberta Edmonton, Alberta T6G 2G1
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper is concernedwith the structure of the arithmetic sum of a finite number of central Cantor sets. The technique used to study this consists of a duality between central Cantor sets and sets of subsums of certain infinite series. One consequence is that the sum of a finite number of central Cantor sets is one of the following: a finite union of closed intervals, homeomorphic to the Cantor ternary set or an $M$-Cantorval.

Keywords

11B05
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 44 , Issue 1 , 01 March 2001 , pp. 12 - 18
Copyright
Copyright © Canadian Mathematical Society 2001

References

[1] Bamon, R., Plaza, S. and Vera, J., On central Cantor sets with self-arithmetic difference of positive measure. J. London Math. Soc. (2) 52 (1995), 137–146.Google Scholar
[2] Carbelli, C. A., Hare, K. E. and Molter, U. M., Sums of Cantor sets. Ergodic Theory Dynamical Systems 17 (1997), 1299–1313.Google Scholar
[3] Guthrie, J. A. and Nymann, J. E., The topological structure of the set of subsums of an infinite series. Coloq. Math. 55 (1988), 323–327.Google Scholar
[4] Kakeya, S., On the partial sums of an infinite series. Tôhoku Sci. Rep. (4) 3 (1914), 159–164.Google Scholar
[5] Mendes, P. and Oliveira, F., On the topological structure of the arithmetic sum of two Cantor sets. Nonlinearity 7 (1994), 329–343.Google Scholar
[6] Menon, P. K., On a class of perfect sets. Bull. Amer.Math. Soc. 54 (1948), 706–711.Google Scholar
[7] Nymann, J. E., Linear combinations of Cantor sets. Coloq. Math. 68 (1995), 259–264.Google Scholar
[8] Palis, J., Homoclinic orbits, hyperbolic dynamics and dimensions of Cantor sets. Contemp. Math. 53(1987).Google Scholar
[9] Sannami, A., An example of a regular Cantor set whose difference set is a Cantor set with positive measure. Hokkaido Math. J. 21 (1992), 7–24.Google Scholar