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arXiv · 2505.03170

The algebraic difference of a Cantor set and its complement

Piotr Nowakowski·Cheng-Han Pan
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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 $\frac{3}{2}$.

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Piotr Nowakowski, Cheng-Han Pan. 2025-05-06. The algebraic difference of a Cantor set and its complement. https://doi.org/10.1090/proc%2F17459

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