Lower bounds for incidences

Let p_1, …, p_n be a collection of points in the unit square and for each i let T_i be a tube through p_i. We prove a lower bound on the number of incidences between these sets of points and tubes under a natural spacing condition. As a corollary, for any collection p_i \in ell_i of n points in the unit square together with a line through each point, there exist j\neq k such that the distance from p_j to ell_k is at most 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 the Heilbronn’s triangle problem attains the high-low limit established in our previous work. Joint work with Alex Cohen and Cosmin Pohoata.