Lower bounds for incidences

Abstract

Let $p_1,\dots ,p_n$ be a set of points in the unit square and let $T_1,\dots ,T_n$ be a set of $\hat{I}´$-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 $p_1,\dots ,p_n\in [0,1{]}^2$ along with a line ${\ell }_j$ through each point $p_j$, there exist $j\ne k$ 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.

53 pages, 2 figures. To appear in Inventiones Mathematicae