Hausdorff dimension of the Apollonian gasket
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913381510807552 |
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| author | Vytnova, Polina Wormell, Caroline |
| author_facet | Vytnova, Polina Wormell, Caroline |
| contents | The Apollonian gasket is a well-studied circle packing. Important properties of the packing, including the distribution of the circle radii, are governed by its Hausdorff dimension. No closed form is currently known for the Hausdorff dimension, and its computation is a special case of a more general and hard problem: effective, rigorous estimates of dimension of a parabolic limit set. In this paper we develop an efficient method for solving this problem which allows us to compute the dimension of the gasket to 128 decimal places and rigorously justify the error bounds. We expect our approach to generalise easily to other parabolic fractals. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_04922 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hausdorff dimension of the Apollonian gasket Vytnova, Polina Wormell, Caroline Dynamical Systems Numerical Analysis Metric Geometry 37F32, 37M22, 37C30, 37D35 The Apollonian gasket is a well-studied circle packing. Important properties of the packing, including the distribution of the circle radii, are governed by its Hausdorff dimension. No closed form is currently known for the Hausdorff dimension, and its computation is a special case of a more general and hard problem: effective, rigorous estimates of dimension of a parabolic limit set. In this paper we develop an efficient method for solving this problem which allows us to compute the dimension of the gasket to 128 decimal places and rigorously justify the error bounds. We expect our approach to generalise easily to other parabolic fractals. |
| title | Hausdorff dimension of the Apollonian gasket |
| topic | Dynamical Systems Numerical Analysis Metric Geometry 37F32, 37M22, 37C30, 37D35 |
| url | https://arxiv.org/abs/2406.04922 |