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Sums of Cantor sets yielding an interval

Part of: Classical measure theory Real functions

Published online by Cambridge University Press:  09 April 2009

Carlos A. Cabrelli ,
Kathryn E. Hare  and
Ursula M. Molter
Carlos A. Cabrelli
Affiliation:
CONICET and, Departmento de Mathematica, FCEyN, Universidad de Buenos Aires, Cdad. Universitaria, Pab. I, (1428) Bs.As, Argentina e-mail: ccabrelli@de. uba. ar
Kathryn E. Hare
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ont. N2L 3G1, Canada e-mail: kehare@math.math.Uwaterloo.ca
Ursula M. Molter
Affiliation:
CONICET and, Departmento de Mathematica, FCEyN, Universidad de Buenos Aires, Cdad. Universitaria, Pab. I, (1428) Bs.As, Argentina e-mail: ccabrelli@de. uba. ar
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Abstract

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In this paper we prove that if a Cantor set has ratios of dissection bounded away from zero, then there is a natural number N, such that its N-fold sum is an interval. Moreover, for each element z of this interval, we explicitly construct the N elements of C whose sum yields z. We also extend a result of Mendes and Oliveria showing that when s is irrational is an interval if and only if a /(1−2a) as/(1−2as) ≥ 1.

Keywords

Cantor setSums of sets

MSC classification

Secondary: 28A80: Fractals 26A30: Singular functions, Cantor functions, functions with other special properties
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 73 , Issue 3 , December 2002 , pp. 405 - 418
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Astels, S., ‘Cantor set numbers with resticted partial quotients’, Trans. Amer. Math. Soc. 352 (2000), 133–170.CrossRefGoogle Scholar
[2]Bamon, R., Plaza, S. and Vera, J., ‘On central Cantor sets with self-arithmetic difference of positive Lebesgue measure’, J. London Math. Soc. 52 (1995), 137–146.CrossRefGoogle Scholar
[3]Brown, G., Keane, M., Moran, W. and Pearce, E., ‘An inequality, with applications to Cantor measures and normal numbers’, Mathematika 35 (1988), 87–94.CrossRefGoogle Scholar
[4]Brown, G. and Moran, W., ‘Raikov systems and radicals in convolution measure algebras’, J. London Math. Soc. 28 (1983), 531–542.CrossRefGoogle Scholar
[5]Cabrelli, C., Hare, K. and Molter, U., ‘Sums of Cantor sets’, Ergodic Theory Dynam. Systems (1997), 1299–1313.CrossRefGoogle Scholar
[6]Hall, M. Jr, ‘On the sum and product of continued fractions’, Ann. of Math. (2) 48 (1947), 966–993.CrossRefGoogle Scholar
[7]Mendes, P. and Oliveira, F., ‘On the topological structure of the arithmetic sum of two Cantor sets’, Nonlinearity 7 (1994), 329–343.CrossRefGoogle Scholar
[8]Moreira, C. and Yoccoz, J., ‘Stable intersections of Cantor sets with large Hausdorff dimension’, Ann. of Math. (2) 154 (2001), 45–96.CrossRefGoogle Scholar
[9]Newhouse, S., ‘Lectures on dynamical systems’, in: Dynamical systems, CIME Lectures, Bressanone, Italy, 1978, Progress in Math. 8 (Birkhauser, Boston, Mass., 1980) pp. 1–114.Google Scholar
[10]Palis, J. and Takens, F., Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations, Cambridge Stud. Adv. Math. 35 (Cambridge University Press, Cambridge, 1993).Google Scholar
[11]Salem, R., ‘On sets of multiplicity for trigonometric series’, Amer J. Math. 64 (1942), 531–538.CrossRefGoogle Scholar
[12]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.CrossRefGoogle Scholar