J. Aust. Math. Soc.  73 (2002), 405-418
Sums of Cantor sets yielding an interval

Carlos A. Cabrelli
  CONICET and
  Departamento de Matematica
  FCEyN, Universidad de Buenos Aires
  Cdad. Universitaria, Pab. I
  (1428) Bs.As.
  Argentina
  ccabrelli@dm.uba.ar
Kathryn E. Hare
  Department of Pure Mathematics
  University of Waterloo
  Waterloo, Ont. N2L 3G1
  Canada
  kehare@math.uwaterloo.ca
and
Ursula M. Molter
  CONICET and
  Departamento de Matematica
  FCEyN, Universidad de Buenos Aires
  Cdad. Universitaria, Pab. I
  (1428) Bs.As.
  Argentina
  umolter@dm.uba.ar


Abstract
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 Oliveira showing that when $s$ is irrational $C_{a}+C_{a^{s}}$ is an interval if and only if $a/(1-2a)\ a^s/(1-2a^s) \geq 1$.
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