When the algebraic difference of two central Cantor sets is an interval?

Abstract

Let \(C(a ),C(b)\subset \lbrack 0,1]\) be the central Cantor sets generated by sequences \(a,b \in \left( 0,1\right)^{\mathbb{N}}\). The first main result of the paper gives a necessary and a sufficient condition for sequences \(a\) and \(b\) which inform when \(C(a )-C(b)\) is equal to \([-1,1]\) or is a finite union of closed intervals. One of the corollaries following from this results shows that the product of thicknesses of two central Cantor sets, the algebraic difference of which is an interval, may be arbitrarily small. We also show that there are sets \(C(a)\) and \(C(b)\) with the Hausdorff dimension equal to 0 such that their algebraic difference is an interval. Finally, we give a full characterization of the case, when \(C(a )-C(b)\) is equal to \([-1,1]\) or is a finite union of closed intervals.