Lots of problems in combinatorics and analysis are connected to incidences: given a set of points and tubes, how much can they intersect? Upper bounds for incidences have been extensively studied, but lower bounds for incidences haven’t received as much attention, and we prove results in this direction. We prove that if you choose n points in the unit square and a line through each point, there is a nontrivial point-line pair with distance <= n^{-2/3+o(1)}. It quickly follows that in any set of n points in the unit square, some three form a triangle of area <= n^{-7/6+o(1)}, a new bound for this problem. The main work is proving a more general incidence lower bound result under a new regularity condition.

Joint with Cosmin Pohoata and Dimitrii Zakharov.