The algebraic difference of central Cantor sets and self-similar Cantor sets

Piotr Nowakowski

Published 2021-02-22Version 1

Let $C(a),C(b)\subset [0,1]$ be the central Cantor sets generated by sequences $a,b \in (0,1)^{\mathbb{N}}$. The main result in the first part of the paper gives a necessary condition 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. In the second part we investigate some self-similar Cantor sets $C(l,r,p)$, which we call S-Cantor sets, generated by numbers $l,r,p \in \mathbb{N}$, $l+r<p$. We give a full characterization of the set $C(l_1,r_1,p)-C(l_2,r_2,p)$ which can take one of the form: the interval $[-1,1]$, a Cantor set, an L-Cantorval, an R-Cantorval or an M-Cantorval.