Badly approximable points on non-linear carpets
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866912963252715520 |
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| author | Anttila, Roope Fraser, Jonathan M. Koivusalo, Henna |
| author_facet | Anttila, Roope Fraser, Jonathan M. Koivusalo, Henna |
| contents | The badly approximable points in $\mathbb{R}^d$ are those for which Dirichlet's approximation theorem cannot be improved by more than a constant, that is, they are the points most difficult to approximate by rational vectors. An important problem in Diophantine approximation is to determine when the set of badly approximable points intersects a given set in full dimension. We find the first class of non-linear non-conformal attractors for which this full intersection property holds, thus answering a question of Das-Fishman-Simmons-Urbański from 2019. We also provide a formula for the Hausdorff dimension of these attractors which is of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_11822 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Badly approximable points on non-linear carpets Anttila, Roope Fraser, Jonathan M. Koivusalo, Henna Number Theory Dynamical Systems Metric Geometry primary: 28A78, 28A80, 11K55, secondary: 37C45 The badly approximable points in $\mathbb{R}^d$ are those for which Dirichlet's approximation theorem cannot be improved by more than a constant, that is, they are the points most difficult to approximate by rational vectors. An important problem in Diophantine approximation is to determine when the set of badly approximable points intersects a given set in full dimension. We find the first class of non-linear non-conformal attractors for which this full intersection property holds, thus answering a question of Das-Fishman-Simmons-Urbański from 2019. We also provide a formula for the Hausdorff dimension of these attractors which is of independent interest. |
| title | Badly approximable points on non-linear carpets |
| topic | Number Theory Dynamical Systems Metric Geometry primary: 28A78, 28A80, 11K55, secondary: 37C45 |
| url | https://arxiv.org/abs/2603.11822 |