Tangency counting for well-spaced circles
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912643692888064 |
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| author | Maldague, Dominique Ortiz, Alexander |
| author_facet | Maldague, Dominique Ortiz, Alexander |
| contents | In the late 90's, Tom Wolff introduced the circle tangency counting problem in his expository article on the Kakeya conjecture. For collections of well-spaced circles, we break the $N^{3/2}$-barrier, proving that a set of $N$ well-spaced circles has at most $N^{25/18+\varepsilon}$ sites of internal tangency. The circle tangency problem can be related to a problem about incidences between points in $\mathbb{R}^3$ and light rays. For this problem, we introduce a stopping time argument to extract maximal information about well-spaced points from a refined decoupling theorem for the light cone in $\mathbb{R}^3$, leading to sharp bounds on the number of $μ$-rich tangency rectangles. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_14118 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tangency counting for well-spaced circles Maldague, Dominique Ortiz, Alexander Classical Analysis and ODEs 42B10, 42B25 In the late 90's, Tom Wolff introduced the circle tangency counting problem in his expository article on the Kakeya conjecture. For collections of well-spaced circles, we break the $N^{3/2}$-barrier, proving that a set of $N$ well-spaced circles has at most $N^{25/18+\varepsilon}$ sites of internal tangency. The circle tangency problem can be related to a problem about incidences between points in $\mathbb{R}^3$ and light rays. For this problem, we introduce a stopping time argument to extract maximal information about well-spaced points from a refined decoupling theorem for the light cone in $\mathbb{R}^3$, leading to sharp bounds on the number of $μ$-rich tangency rectangles. |
| title | Tangency counting for well-spaced circles |
| topic | Classical Analysis and ODEs 42B10, 42B25 |
| url | https://arxiv.org/abs/2504.14118 |