Packing sets in Euclidean space by affine transformations
| Autor: | Iosevich, Alex, Mattila, Pertti, Palsson, Eyvindur, Pham, Minh-Quy, Pham, Thang, Senger, Steven, Shen, Chun-Yen |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | For Borel subsets $\Theta\subset O(d)\times \mathbb{R}^d$ (the set of all rigid motions) and $E\subset \mathbb{R}^d$, we define \begin{align*} \Theta(E):=\bigcup_{(g,z)\in \Theta}(gE+z). \end{align*} In this paper, we investigate the Lebesgue measure and Hausdorff dimension of $\Theta(E)$ given the dimensions of the Borel sets $E$ and $\Theta$, when $\Theta$ 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. Comment: 27 pages |
| Databáze: | arXiv |
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