Interior of certain sums and continuous images of very thin Cantor sets
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| Format: | Preprint |
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2024
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| _version_ | 1866909332930560000 |
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| author | Jung, Yeonwook Lai, Chun-Kit |
| author_facet | Jung, Yeonwook Lai, Chun-Kit |
| contents | We show that for all Cantor set $K_1$ on ${\mathbb R}^d$, it is always possible to find another Cantor set $K_2$ so that the sum $g(K_1)+ K_2$ (where $g$ is a $C^1$ local diffeomorphism) has non-empty interior, and the existence of the interior is robust under small perturbation of the mapping. More generally, we can also show that the image set $H(α, K_1,K_2)$, where $H$ is some $C^1$ function on ${\mathbb R}^N\times{\mathbb R}^d\times{\mathbb R}^d$ with non-vanishing Jacobian, have non-empty interior for $α$ all in an open ball of ${\mathbb R}^N$. This result allows us to show that all Cantor sets are not topologically universal using $C^1$ local diffeomorphism, proving a stronger version of the topological Erdős similarity conjecture. Moreover, we are also able to construct a Cantor set of dimension $d$ on ${\mathbb R}^{2d}$, whose distance set has an interior. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_01267 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Interior of certain sums and continuous images of very thin Cantor sets Jung, Yeonwook Lai, Chun-Kit Metric Geometry Classical Analysis and ODEs Dynamical Systems We show that for all Cantor set $K_1$ on ${\mathbb R}^d$, it is always possible to find another Cantor set $K_2$ so that the sum $g(K_1)+ K_2$ (where $g$ is a $C^1$ local diffeomorphism) has non-empty interior, and the existence of the interior is robust under small perturbation of the mapping. More generally, we can also show that the image set $H(α, K_1,K_2)$, where $H$ is some $C^1$ function on ${\mathbb R}^N\times{\mathbb R}^d\times{\mathbb R}^d$ with non-vanishing Jacobian, have non-empty interior for $α$ all in an open ball of ${\mathbb R}^N$. This result allows us to show that all Cantor sets are not topologically universal using $C^1$ local diffeomorphism, proving a stronger version of the topological Erdős similarity conjecture. Moreover, we are also able to construct a Cantor set of dimension $d$ on ${\mathbb R}^{2d}$, whose distance set has an interior. |
| title | Interior of certain sums and continuous images of very thin Cantor sets |
| topic | Metric Geometry Classical Analysis and ODEs Dynamical Systems |
| url | https://arxiv.org/abs/2410.01267 |