A note on limsup sets of annuli
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915152080666624 |
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| author | Hussain, Mumtaz Ward, Benjamin |
| author_facet | Hussain, Mumtaz Ward, Benjamin |
| contents | We consider the set of points in infinitely many max-norm annuli centred at rational points in $\mathbb R^{n}$. We give Jarník-Besicovitch type theorems for this set in terms of Hausdorff dimension. Interestingly, we find that if the outer radii are decreasing sufficiently slowly, dependent only on the dimension $n$, and the thickness of the annuli is decreasing rapidly then the dimension of the set tends towards $n-1$. We also consider various other forms of annuli including rectangular annuli and quasi-annuli described by the difference between balls of two different norms. Our results are deduced through a novel combination of a version of Cassel's Scaling Lemma and a generalisation of the Mass Transference Principle, namely the Mass transference principle from rectangles to rectangles due to Wang and Wu (Math. Ann. 2021). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09807 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on limsup sets of annuli Hussain, Mumtaz Ward, Benjamin Number Theory Dynamical Systems Metric Geometry We consider the set of points in infinitely many max-norm annuli centred at rational points in $\mathbb R^{n}$. We give Jarník-Besicovitch type theorems for this set in terms of Hausdorff dimension. Interestingly, we find that if the outer radii are decreasing sufficiently slowly, dependent only on the dimension $n$, and the thickness of the annuli is decreasing rapidly then the dimension of the set tends towards $n-1$. We also consider various other forms of annuli including rectangular annuli and quasi-annuli described by the difference between balls of two different norms. Our results are deduced through a novel combination of a version of Cassel's Scaling Lemma and a generalisation of the Mass Transference Principle, namely the Mass transference principle from rectangles to rectangles due to Wang and Wu (Math. Ann. 2021). |
| title | A note on limsup sets of annuli |
| topic | Number Theory Dynamical Systems Metric Geometry |
| url | https://arxiv.org/abs/2502.09807 |