Fractal Sumset Properties
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910470018957312 |
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| author | Kong, Derong Wang, Zhiqiang |
| author_facet | Kong, Derong Wang, Zhiqiang |
| contents | In this paper we introduce two notions of fractal sumset properties. A compact set $K\subset\mathbb{R}^d$ is said to have the Hausdorff sumset property (HSP) if for any $\ell\in\mathbb{N}_{\ge 2}$ there exist compact sets $K_1, K_2,\ldots, K_\ell$ such that $K_1+K_2+\cdots+K_\ell\subset K$ and $\dim_H K_i=\dim_H K$ for all $1\le i\le \ell$. Analogously, if we replace the Hausdorff dimension by the packing dimension in the definition of HSP, then the compact set $K\subset\mathbb{R}^d$ is said to have the packing sumset property (PSP). We show that the HSP fails for certain homogeneous self-similar sets satisfying the strong separation condition, while the PSP holds for all homogeneous self-similar sets in $\mathbb{R}^d$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_10404 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Fractal Sumset Properties Kong, Derong Wang, Zhiqiang Classical Analysis and ODEs Primary: 28A80, Secondary: 11B13, 28A78 In this paper we introduce two notions of fractal sumset properties. A compact set $K\subset\mathbb{R}^d$ is said to have the Hausdorff sumset property (HSP) if for any $\ell\in\mathbb{N}_{\ge 2}$ there exist compact sets $K_1, K_2,\ldots, K_\ell$ such that $K_1+K_2+\cdots+K_\ell\subset K$ and $\dim_H K_i=\dim_H K$ for all $1\le i\le \ell$. Analogously, if we replace the Hausdorff dimension by the packing dimension in the definition of HSP, then the compact set $K\subset\mathbb{R}^d$ is said to have the packing sumset property (PSP). We show that the HSP fails for certain homogeneous self-similar sets satisfying the strong separation condition, while the PSP holds for all homogeneous self-similar sets in $\mathbb{R}^d$. |
| title | Fractal Sumset Properties |
| topic | Classical Analysis and ODEs Primary: 28A80, Secondary: 11B13, 28A78 |
| url | https://arxiv.org/abs/2308.10404 |