The algebraic difference of a Cantor set and its complement
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914410537156608 |
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| author | Nowakowski, Piotr Pan, Cheng-Han |
| author_facet | Nowakowski, Piotr Pan, Cheng-Han |
| contents | 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 $\frac{3}{2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_03170 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The algebraic difference of a Cantor set and its complement Nowakowski, Piotr Pan, Cheng-Han Classical Analysis and ODEs 28A05, 28A80 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 $\frac{3}{2}$. |
| title | The algebraic difference of a Cantor set and its complement |
| topic | Classical Analysis and ODEs 28A05, 28A80 |
| url | https://arxiv.org/abs/2505.03170 |