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 $\delta $ -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.
