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 havenâ€™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.