Radial Projections in $\mathbb{R}^n$ Revisited
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929386505109504 |
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| author | Bright, Paige Fu, Yuqiu Ren, Kevin |
| author_facet | Bright, Paige Fu, Yuqiu Ren, Kevin |
| contents | We generalize the recent results on radial projections by Orponen, Shmerkin, Wang using two different methods. In particular, we show that given $X,Y\subset \mathbb{R}^n$ Borel sets and $X\neq \emptyset$. If $\dim Y \in (k,k+1]$ for some $k\in \{1,\dots, n-1\}$, then \[ \sup_{x\in X} \dim π_x(Y\setminus \{x\}) \geq \min \{\dim X + \dim Y - k, k\}. \] Our results give a new approach to solving a conjecture of Lund-Pham-Thu in all dimensions and for all ranges of $\dim Y$.
The first of our two methods for proving the above theorem is shorter, utilizing a result of the first author and Gan. Our second method, though longer, follows the original methodology of Orponen--Shmerkin--Wang, and requires a higher dimensional incidence estimate and a dual Furstenberg-set estimate for lines. These new estimates may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09707 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Radial Projections in $\mathbb{R}^n$ Revisited Bright, Paige Fu, Yuqiu Ren, Kevin Classical Analysis and ODEs Combinatorics Metric Geometry 28A78, 28A80 We generalize the recent results on radial projections by Orponen, Shmerkin, Wang using two different methods. In particular, we show that given $X,Y\subset \mathbb{R}^n$ Borel sets and $X\neq \emptyset$. If $\dim Y \in (k,k+1]$ for some $k\in \{1,\dots, n-1\}$, then \[ \sup_{x\in X} \dim π_x(Y\setminus \{x\}) \geq \min \{\dim X + \dim Y - k, k\}. \] Our results give a new approach to solving a conjecture of Lund-Pham-Thu in all dimensions and for all ranges of $\dim Y$. The first of our two methods for proving the above theorem is shorter, utilizing a result of the first author and Gan. Our second method, though longer, follows the original methodology of Orponen--Shmerkin--Wang, and requires a higher dimensional incidence estimate and a dual Furstenberg-set estimate for lines. These new estimates may be of independent interest. |
| title | Radial Projections in $\mathbb{R}^n$ Revisited |
| topic | Classical Analysis and ODEs Combinatorics Metric Geometry 28A78, 28A80 |
| url | https://arxiv.org/abs/2406.09707 |