Sums of Cantor sets yielding an interval

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 Ca + Cas is an interval if and only if a/(1 - 2a) as/(1 - 2as) ≥ 1.