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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2410.19237 |
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| _version_ | 1866915095417716736 |
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| author | MacVicar, Neil |
| author_facet | MacVicar, Neil |
| contents | Let $C$ be the attractor of the IFS $\{f_{d}(z) = (-n+i)^{-1}(z+d): d\in D\}$, $D\subset\{0, 1, \ldots, n^{2}\}$ and let $\dim$ denote the box-counting dimension. It is known that for all $λ\in[0, 1]$, that the set of complex numbers $α$ for which $\dim(C\cap(C+α)) = λ\dim(C)$ is dense in the set of $α$ for which $C \cap (C + α) \neq \emptyset$ when $d \leq n^{2}/2$ for all $d\in D$ and $|δ- δ^{'}| > n$ for all $δ\neq δ^{'} \in D - D$. We show that this result still holds when we replace $|δ- δ^{'}| > n$ with $|δ- δ^{'}| > 1$. In fact, for sufficiently large $n$, the result even holds when we remove the assumption $d\leq n^{2}/2$ and replace $|δ- δ^{'}| > n$ by $|δ- δ^{'}| > 2$. Additionally, we make similar statements where $\dim$ denotes the Hausdorff dimension or packing dimension. Our insights also find application in classifying the self-similarity of $C\cap(C+α)$. Namely we connect the occurrence of self-similarity to the notion of strongly eventually periodic sequences seen for analogous objects on the real line. We also provide a new proof of a result of W. Gilbert that inspired this work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_19237 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Intersections of Cantor Sets Derived from Complex Radix Expansions MacVicar, Neil Dynamical Systems 28A80, 11A63 Let $C$ be the attractor of the IFS $\{f_{d}(z) = (-n+i)^{-1}(z+d): d\in D\}$, $D\subset\{0, 1, \ldots, n^{2}\}$ and let $\dim$ denote the box-counting dimension. It is known that for all $λ\in[0, 1]$, that the set of complex numbers $α$ for which $\dim(C\cap(C+α)) = λ\dim(C)$ is dense in the set of $α$ for which $C \cap (C + α) \neq \emptyset$ when $d \leq n^{2}/2$ for all $d\in D$ and $|δ- δ^{'}| > n$ for all $δ\neq δ^{'} \in D - D$. We show that this result still holds when we replace $|δ- δ^{'}| > n$ with $|δ- δ^{'}| > 1$. In fact, for sufficiently large $n$, the result even holds when we remove the assumption $d\leq n^{2}/2$ and replace $|δ- δ^{'}| > n$ by $|δ- δ^{'}| > 2$. Additionally, we make similar statements where $\dim$ denotes the Hausdorff dimension or packing dimension. Our insights also find application in classifying the self-similarity of $C\cap(C+α)$. Namely we connect the occurrence of self-similarity to the notion of strongly eventually periodic sequences seen for analogous objects on the real line. We also provide a new proof of a result of W. Gilbert that inspired this work. |
| title | Intersections of Cantor Sets Derived from Complex Radix Expansions |
| topic | Dynamical Systems 28A80, 11A63 |
| url | https://arxiv.org/abs/2410.19237 |