The algebraic difference of a Cantor set and its complement

TL;DR

This paper investigates the algebraic difference between a Cantor set and its complement, revealing that the Lebesgue measure of this difference has a greatest lower bound of 1.5, highlighting a delicate balance in their sizes.

Contribution

It establishes a novel lower bound for the Lebesgue measure of the difference between a Cantor set's complement and the set itself.

Findings

Lebesgue measure of $ ext{C}^c - ext{C}$ has a greatest lower bound of 3/2.

Explores the interplay between a Cantor set and its complement in measure theory.

Highlights the delicate balance affecting the size of the difference set.

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 $\frac{3}{2}$.