تقرير
Large cliques in extremal incidence configurations
| العنوان: | Large cliques in extremal incidence configurations |
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| المؤلفون: | Orponen, Tuomas, Yi, Guangzeng |
| سنة النشر: | 2024 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Combinatorics, Mathematics - Classical Analysis and ODEs, 28A80 (primary), 05B99, 05D99, 51A20 (secondary) |
| الوصف: | Let $P \subset \mathbb{R}^{2}$ be a Katz-Tao $(\delta,s)$-set, and let $\mathcal{L}$ be a Katz-Tao $(\delta,t)$-set of lines in $\mathbb{R}^{2}$. A recent result of Fu and Ren gives a sharp upper bound for the $\delta$-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 $s,t \in (0,1]$, $$ |\mathcal{I}(P,\mathcal{L})|_{\delta} \lesssim_{\epsilon} \delta^{-\epsilon -f(s,t)}, \qquad \epsilon > 0,$$ 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 $|\mathcal{I}(P,\mathcal{L})|_{\delta} \approx \delta^{-f(s,t)}$, then $P \times \mathcal{L}$ contains "cliques" $P' \times \mathcal{L}'$ satisfying $|\mathcal{I}(P',\mathcal{L}')|_{\delta} \approx |P'|_{\delta}|\mathcal{L}'|_{\delta}$, $$|P'|_{\delta} \approx \delta^{-s^{2}/(s + t)} \quad \text{and} \quad |\mathcal{L}'|_{\delta} \approx \delta^{-t^{2}/(s + t)}.$$ Comment: 27 pages, 2 figures |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2402.12104 |
| رقم الأكسشن: | edsarx.2402.12104 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |