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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.05808 |
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| _version_ | 1866918089862414336 |
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| author | Cai, Yi Yang, Yang |
| author_facet | Cai, Yi Yang, Yang |
| contents | The visible problem is related to the arithmetic on the fractals. The visibility of self-similar set has been studied in the past. In this work, we investigate the visibility of non-self-similar sets. We begin by analyzing the structure of $F^2_λ$, where $F^2_λ:=\set{x^2:x\in F_λ}$ and $F_λ$ is the middle $1-2λ$ Cantor set, we show that it lacks self-similarity. Due to the nonlinear phenomena exhibited by $F^2_λ$, we develop a different approach to characterize the visible set. %combining methods from fractal theory, numerical computation, and dynamical systems theory. Our results also reveal that the visible set may contain a closed interval within a large range of $λ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_05808 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Visibility of non-self-similar sets Cai, Yi Yang, Yang Number Theory The visible problem is related to the arithmetic on the fractals. The visibility of self-similar set has been studied in the past. In this work, we investigate the visibility of non-self-similar sets. We begin by analyzing the structure of $F^2_λ$, where $F^2_λ:=\set{x^2:x\in F_λ}$ and $F_λ$ is the middle $1-2λ$ Cantor set, we show that it lacks self-similarity. Due to the nonlinear phenomena exhibited by $F^2_λ$, we develop a different approach to characterize the visible set. %combining methods from fractal theory, numerical computation, and dynamical systems theory. Our results also reveal that the visible set may contain a closed interval within a large range of $λ$. |
| title | Visibility of non-self-similar sets |
| topic | Number Theory |
| url | https://arxiv.org/abs/2505.05808 |