Large cliques in extremal incidence configurations

Abstract

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 $s,t\in (0,1]$, $$|\mathcal{I}(P,\mathcal{L}){|}_{\delta }{\lesssim }_{\epsilon }{\delta }^{-\epsilon -f(s,t)},\epsilon >0,$$ 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" $P^{\prime }\times {\mathcal{L}}^{\prime }$ satisfying $|\mathcal{I}(P^{\prime },{\mathcal{L}}^{\prime }){|}_{\delta }\approx |P^{\prime }{|}_{\delta }|{\mathcal{L}}^{\prime }{|}_{\delta }$, $$|P^{\prime }{|}_{\delta }\approx {\delta }^{-s^2/(s+t)}\text{and}|{\mathcal{L}}^{\prime }{|}_{\delta }\approx {\delta }^{-t^2/(s+t)}.$$

27 pages, 2 figures