Characterization of the algebraic difference of special affine Cantor sets

TL;DR

This paper characterizes the algebraic difference of certain self-similar Cantor sets, revealing when the difference forms intervals, Cantor sets, or Cantorvals, and provides examples using positional numeral systems.

Contribution

It provides a complete characterization of the difference sets of special affine Cantor sets, including conditions for various set types and new examples via numeral systems.

Findings

The difference can be an interval, Cantor set, or Cantorval.

Explicit conditions for the set types based on parameters.

Examples of sets described through positional numeral systems.

Abstract

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. As corollaries we give examples of Cantor sets and Cantorvals, which can be easily described using some positional numeral systems.