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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.19986 |
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| _version_ | 1866911613915758592 |
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| author | MacVicar, Neil |
| author_facet | MacVicar, Neil |
| contents | This thesis generalizes the study of $C\cap(C + α)$ where $C$ is the middle third Cantor set to self-affine sets in $\mathbb{R}^{n}$. We present sufficient and necessary conditions for when the translation $α$ produces a self-affine intersection for a particular class of self-affine sets. In the case where the attractor is self-similar, we improve results concerning the function from $α$ to the fractal dimension of the intersection. This lends itself to a case study of the complex number system $(-n + i, \{0, 1, . . . , n^{2}\})$, when $n$ is an integer greater than or equal to $2$. Lastly, we present a definition of multiplicative invariance for subsets of $\mathbb{Z}^{n}$ and establish a connection, known in the one-dimensional case, between them and invariant sets of the $n$-dimensional torus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19986 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the intersections of homogeneous self-similar sets with their translates in $\mathbb{R}^{n}$ and a formulation of multiplicative invariance in $\mathbb{Z}^{n}$ MacVicar, Neil Dynamical Systems Metric Geometry 28A80, 11A63 This thesis generalizes the study of $C\cap(C + α)$ where $C$ is the middle third Cantor set to self-affine sets in $\mathbb{R}^{n}$. We present sufficient and necessary conditions for when the translation $α$ produces a self-affine intersection for a particular class of self-affine sets. In the case where the attractor is self-similar, we improve results concerning the function from $α$ to the fractal dimension of the intersection. This lends itself to a case study of the complex number system $(-n + i, \{0, 1, . . . , n^{2}\})$, when $n$ is an integer greater than or equal to $2$. Lastly, we present a definition of multiplicative invariance for subsets of $\mathbb{Z}^{n}$ and establish a connection, known in the one-dimensional case, between them and invariant sets of the $n$-dimensional torus. |
| title | On the intersections of homogeneous self-similar sets with their translates in $\mathbb{R}^{n}$ and a formulation of multiplicative invariance in $\mathbb{Z}^{n}$ |
| topic | Dynamical Systems Metric Geometry 28A80, 11A63 |
| url | https://arxiv.org/abs/2604.19986 |