Arithmetic structure of the exceptional set of projections
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866917837924204544 |
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| author | Chen, Changhao Miao, Zhengyan |
| author_facet | Chen, Changhao Miao, Zhengyan |
| contents | We study the arithmetic structure of the exceptional set of projections. For any bounded subset $E\subset \mathbb{R}^d$, let $$ Ω=\{ξ\in \mathbb{R}: \dim_B(E+ξE)=\dim_B E\}. $$ We prove that either $Ω=\{0\}$ or $Ω$ is a subfield of $\mathbb{R}$. We show that in general the statement does not hold for Hausdorff dimension and lower box dimension. Moreover, for any $s\in (0, 1]$ and a sequence $(r_k) \subset \mathbb{R}$, we construct a Ahlfors $s$-regular set $E\subset \mathbb{R}^2$ such that for any $r_k, k\in \mathbb{N}$, we have \[ \overline{\dim}_B \, \{x+r_k\, y: (x, y)\in E\} <s. \] |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_12677 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Arithmetic structure of the exceptional set of projections Chen, Changhao Miao, Zhengyan Classical Analysis and ODEs We study the arithmetic structure of the exceptional set of projections. For any bounded subset $E\subset \mathbb{R}^d$, let $$ Ω=\{ξ\in \mathbb{R}: \dim_B(E+ξE)=\dim_B E\}. $$ We prove that either $Ω=\{0\}$ or $Ω$ is a subfield of $\mathbb{R}$. We show that in general the statement does not hold for Hausdorff dimension and lower box dimension. Moreover, for any $s\in (0, 1]$ and a sequence $(r_k) \subset \mathbb{R}$, we construct a Ahlfors $s$-regular set $E\subset \mathbb{R}^2$ such that for any $r_k, k\in \mathbb{N}$, we have \[ \overline{\dim}_B \, \{x+r_k\, y: (x, y)\in E\} <s. \] |
| title | Arithmetic structure of the exceptional set of projections |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2210.12677 |