Large cliques in extremal incidence configurations

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_{\varepsilon}\delta^{-\varepsilon -f(s,t)}, \quad \varepsilon > 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)}.