Lower bounds for incidences

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Lots of problems in combinatorics and analysis are connected to upper bounds for incidences: given a set of points and tubes, how much can they intersect? On the other hand, lower bounds for incidences have not been studied much. In this vein, we prove that if you choose `n’ points in the unit square and a line through each point, then 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 work with Cosmin Pohoata and Dmitrii Zakharov.

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