A Note on Cantor Sets

Abstract

The Cantor set is constructed by the iterate deletion of a middle interval equidistant from the end points. It is well known that the sums of points in the set cover completely the real line. It was an open problem to know if this property was still true for the sets obtained when the deleted interval is not any more equidistant from the end points. In this note we answer this question positively. We give a simple proof that reflects the geometric nature of the problem, and that is a variation on an old idea that goes back to Steinhaus.