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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2402.12104 |
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| _version_ | 1866914685092102144 |
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| author | Orponen, Tuomas Yi, Guangzeng |
| author_facet | Orponen, Tuomas Yi, Guangzeng |
| contents | Let $P \subset \mathbb{R}^{2}$ be a Katz-Tao $(δ,s)$-set, and let $\mathcal{L}$ be a Katz-Tao $(δ,t)$-set of lines in $\mathbb{R}^{2}$. A recent result of Fu and Ren gives a sharp upper bound for the $δ$-covering number of the set of incidences $\mathcal{I}(P,\mathcal{L}) = \{(p,\ell) \in P \times \mathcal{L} : p \in \ell\}$. In fact, for $s,t \in (0,1]$, $$ |\mathcal{I}(P,\mathcal{L})|_δ \lesssim_ε δ^{-ε-f(s,t)}, \qquad ε> 0,$$ where $f(s,t) = (s^{2} + st + t^{2})/(s + t)$. For $s,t \in (0,1]$, we characterise the near-extremal configurations $P \times \mathcal{L}$ of this inequality: we show that if $|\mathcal{I}(P,\mathcal{L})|_δ \approx δ^{-f(s,t)}$, then $P \times \mathcal{L}$ contains "cliques" $P' \times \mathcal{L}'$ satisfying $|\mathcal{I}(P',\mathcal{L}')|_δ \approx |P'|_δ|\mathcal{L}'|_δ$, $$|P'|_δ \approx δ^{-s^{2}/(s + t)} \quad \text{and} \quad |\mathcal{L}'|_δ \approx δ^{-t^{2}/(s + t)}.$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12104 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Large cliques in extremal incidence configurations Orponen, Tuomas Yi, Guangzeng Combinatorics Classical Analysis and ODEs 28A80 (primary), 05B99, 05D99, 51A20 (secondary) Let $P \subset \mathbb{R}^{2}$ be a Katz-Tao $(δ,s)$-set, and let $\mathcal{L}$ be a Katz-Tao $(δ,t)$-set of lines in $\mathbb{R}^{2}$. A recent result of Fu and Ren gives a sharp upper bound for the $δ$-covering number of the set of incidences $\mathcal{I}(P,\mathcal{L}) = \{(p,\ell) \in P \times \mathcal{L} : p \in \ell\}$. In fact, for $s,t \in (0,1]$, $$ |\mathcal{I}(P,\mathcal{L})|_δ \lesssim_ε δ^{-ε-f(s,t)}, \qquad ε> 0,$$ where $f(s,t) = (s^{2} + st + t^{2})/(s + t)$. For $s,t \in (0,1]$, we characterise the near-extremal configurations $P \times \mathcal{L}$ of this inequality: we show that if $|\mathcal{I}(P,\mathcal{L})|_δ \approx δ^{-f(s,t)}$, then $P \times \mathcal{L}$ contains "cliques" $P' \times \mathcal{L}'$ satisfying $|\mathcal{I}(P',\mathcal{L}')|_δ \approx |P'|_δ|\mathcal{L}'|_δ$, $$|P'|_δ \approx δ^{-s^{2}/(s + t)} \quad \text{and} \quad |\mathcal{L}'|_δ \approx δ^{-t^{2}/(s + t)}.$$ |
| title | Large cliques in extremal incidence configurations |
| topic | Combinatorics Classical Analysis and ODEs 28A80 (primary), 05B99, 05D99, 51A20 (secondary) |
| url | https://arxiv.org/abs/2402.12104 |