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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2411.14861 |
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| _version_ | 1866917845077590016 |
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| author | Pourbarat, Mehdi |
| author_facet | Pourbarat, Mehdi |
| contents | Suppose that $K$ and $ K'$ are two affine Cantor sets. It is shown that the sum set $K+K'$ has equal box and Hausdorff dimensions and in this number named $s$, $H^s(K+K')<\infty$. Moreover, for almost every pair $(K,K')$ satisfying $HD(K)+HD(K')\leq 1$, there is a dense subset $D\subset \mathbb R$ such that $H^s(K+λK')=0$, for all $λ\in D$. It also is shown that in the context of affine Cantor sets with two increasing maps, there are generically (topological and almost everywhere) five possible structures for their sum: a Cantor set, an L, R, M-Cantorval or a finite union of closed intervals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_14861 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the sum of two affine Cantor sets Pourbarat, Mehdi Dynamical Systems 28A78, 58F14 Suppose that $K$ and $ K'$ are two affine Cantor sets. It is shown that the sum set $K+K'$ has equal box and Hausdorff dimensions and in this number named $s$, $H^s(K+K')<\infty$. Moreover, for almost every pair $(K,K')$ satisfying $HD(K)+HD(K')\leq 1$, there is a dense subset $D\subset \mathbb R$ such that $H^s(K+λK')=0$, for all $λ\in D$. It also is shown that in the context of affine Cantor sets with two increasing maps, there are generically (topological and almost everywhere) five possible structures for their sum: a Cantor set, an L, R, M-Cantorval or a finite union of closed intervals. |
| title | On the sum of two affine Cantor sets |
| topic | Dynamical Systems 28A78, 58F14 |
| url | https://arxiv.org/abs/2411.14861 |