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| Main Authors: | , , , , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.03087 |
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| _version_ | 1866910436156243968 |
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| author | Iosevich, Alex Mattila, Pertti Palsson, Eyvindur Pham, Minh-Quy Pham, Thang Senger, Steven Shen, Chun-Yen |
| author_facet | Iosevich, Alex Mattila, Pertti Palsson, Eyvindur Pham, Minh-Quy Pham, Thang Senger, Steven Shen, Chun-Yen |
| contents | For Borel subsets $Θ\subset O(d)\times \mathbb{R}^d$ (the set of all rigid motions) and $E\subset \mathbb{R}^d$, we define \begin{align*}
Θ(E):=\bigcup_{(g,z)\in Θ}(gE+z). \end{align*} In this paper, we investigate the Lebesgue measure and Hausdorff dimension of $Θ(E)$ given the dimensions of the Borel sets $E$ and $Θ$, when $Θ$ has product form. We also study this question by replacing rigid motions with the class of dilations and translations; and similarity transformations. The dimensional thresholds are sharp. Our results are variants of some previously known results in the literature when $E$ is restricted to smooth objects such as spheres, $k$-planes, and surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03087 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Packing sets in Euclidean space by affine transformations Iosevich, Alex Mattila, Pertti Palsson, Eyvindur Pham, Minh-Quy Pham, Thang Senger, Steven Shen, Chun-Yen Classical Analysis and ODEs 28A75 For Borel subsets $Θ\subset O(d)\times \mathbb{R}^d$ (the set of all rigid motions) and $E\subset \mathbb{R}^d$, we define \begin{align*} Θ(E):=\bigcup_{(g,z)\in Θ}(gE+z). \end{align*} In this paper, we investigate the Lebesgue measure and Hausdorff dimension of $Θ(E)$ given the dimensions of the Borel sets $E$ and $Θ$, when $Θ$ has product form. We also study this question by replacing rigid motions with the class of dilations and translations; and similarity transformations. The dimensional thresholds are sharp. Our results are variants of some previously known results in the literature when $E$ is restricted to smooth objects such as spheres, $k$-planes, and surfaces. |
| title | Packing sets in Euclidean space by affine transformations |
| topic | Classical Analysis and ODEs 28A75 |
| url | https://arxiv.org/abs/2405.03087 |