Lower bounds for incidences
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915202552823808 |
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| author | Cohen, Alex Pohoata, Cosmin Zakharov, Dmitrii |
| author_facet | Cohen, Alex Pohoata, Cosmin Zakharov, Dmitrii |
| contents | Let $p_1,\ldots,p_n$ be a set of points in the unit square and let $T_1,\ldots,T_n$ be a set of $δ$-tubes such that $T_j$ passes through $p_j$. We prove a lower bound for the number of incidences between the points and tubes under a natural regularity condition (similar to Frostman regularity). As a consequence, we show that in any configuration of points $p_1,\ldots, p_n \in [0,1]^2$ along with a line $\ell_j$ through each point $p_j$, there exist $j\neq k$ for which $d(p_j, \ell_k) \lesssim n^{-2/3+o(1)}$.
It follows from the latter result that any set of $n$ points in the unit square contains three points forming a triangle of area at most $n^{-7/6+o(1)}$. This new upper bound for Heilbronn's triangle problem attains the high-low limit established in our previous work arXiv:2305.18253. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_07658 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lower bounds for incidences Cohen, Alex Pohoata, Cosmin Zakharov, Dmitrii Combinatorics Classical Analysis and ODEs Metric Geometry Let $p_1,\ldots,p_n$ be a set of points in the unit square and let $T_1,\ldots,T_n$ be a set of $δ$-tubes such that $T_j$ passes through $p_j$. We prove a lower bound for the number of incidences between the points and tubes under a natural regularity condition (similar to Frostman regularity). As a consequence, we show that in any configuration of points $p_1,\ldots, p_n \in [0,1]^2$ along with a line $\ell_j$ through each point $p_j$, there exist $j\neq k$ for which $d(p_j, \ell_k) \lesssim n^{-2/3+o(1)}$. It follows from the latter result that any set of $n$ points in the unit square contains three points forming a triangle of area at most $n^{-7/6+o(1)}$. This new upper bound for Heilbronn's triangle problem attains the high-low limit established in our previous work arXiv:2305.18253. |
| title | Lower bounds for incidences |
| topic | Combinatorics Classical Analysis and ODEs Metric Geometry |
| url | https://arxiv.org/abs/2409.07658 |