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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

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