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

TL;DR

This paper establishes conditions under which the algebraic difference of two central Cantor sets is an interval or a finite union of intervals, revealing surprising cases with small thickness product and zero Hausdorff dimension.

Contribution

It provides a complete characterization of when the difference of two central Cantor sets forms an interval or union of intervals, including cases with minimal thickness and zero dimension.

Findings

Necessary and sufficient conditions for the difference to be an interval

Existence of zero-dimensional sets with interval difference

Interval difference can occur with arbitrarily small thickness product

Abstract

Let $C(a ),C(b)\subset \lbrack 0,1]$ be the central Cantor sets generated by sequences $ a,b \in (0,1)^{\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 which algebraic difference 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.