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Sums of Cantor sets yielding an interval

Published online by Cambridge University Press:  09 April 2009

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: [email protected]
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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2002

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