The algebraic difference of a Cantor set and its complement

Abstract

Let $\mathcal{C}\subseteq [0,1]$ be a Cantor set. In the classical $\mathcal{C}\pm \mathcal{C}$ problems, modifying the ``size'' of $\mathcal{C}$ has a magnified effect on $\mathcal{C}\pm \mathcal{C}$. However, any gain in $\mathcal{C}$ necessarily results in a loss in ${\mathcal{C}}^c$, and vice versa. This interplay between $\mathcal{C}$ and its complement ${\mathcal{C}}^c$ raises interesting questions about the delicate balance between the two, particularly in how it influences the ``size'' of ${\mathcal{C}}^c-\mathcal{C}$. One of our main results indicates that the Lebesgue measure of ${\mathcal{C}}^c-\mathcal{C}$ has a greatest lower bound of $\frac{3}{2}$.