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

Abstract

Let $C(a),C(b)\subset [0,1]$ 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.

Final published version available on: https://afm.journal.fi/article/view/126014

References in corpus (1)

• Thickness theorems with partial derivatives

Cited by in corpus (3)

• The achievement set of generalized multigeometric sequences
• Conditions for the difference set of a central Cantor set to be a Cantorval. Part II
• The algebraic difference of a Cantor set and its complement