Topological, metric and fractal properties of one family of self-similar sets
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915843310354432 |
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| author | Karvatskyi, Dmytro |
| author_facet | Karvatskyi, Dmytro |
| contents | Depending on a natural parameter $l$, we study the topological, metric, and fractal properties of the homogeneous self-similar set $$K_{l}=\left\{\sum_{i=1}^{\infty} \frac{\varepsilon_i}{(2l+2)^i} : (\varepsilon_i) \in \{0, 2, 4, \dots, 2l, 2l+1, 2l+3, \dots, 4l+1 \}^{\mathbb{N}} \right\}.$$ In particular, we prove that $K_l$ is a Cantorval, that is, a perfect set on the real line with a non-empty interior and fractal boundary. Additionally, we compute the Lebesgue measure of $K_l$ and the Hausdorff dimension of its boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_07575 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Topological, metric and fractal properties of one family of self-similar sets Karvatskyi, Dmytro Dynamical Systems 28A80, 28A78, 37B10 Depending on a natural parameter $l$, we study the topological, metric, and fractal properties of the homogeneous self-similar set $$K_{l}=\left\{\sum_{i=1}^{\infty} \frac{\varepsilon_i}{(2l+2)^i} : (\varepsilon_i) \in \{0, 2, 4, \dots, 2l, 2l+1, 2l+3, \dots, 4l+1 \}^{\mathbb{N}} \right\}.$$ In particular, we prove that $K_l$ is a Cantorval, that is, a perfect set on the real line with a non-empty interior and fractal boundary. Additionally, we compute the Lebesgue measure of $K_l$ and the Hausdorff dimension of its boundary. |
| title | Topological, metric and fractal properties of one family of self-similar sets |
| topic | Dynamical Systems 28A80, 28A78, 37B10 |
| url | https://arxiv.org/abs/2603.07575 |