Analysis Seminar

Incidence lower bounds for tubes

Location: Warren Weaver Hall 1302

Date: Thursday, November 7, 2024, 11 a.m.

Synopsis:

There have been lots of works proving upper bounds on the number of incidences between points and tubes, with many applications to analysis and combinatorics. We consider the opposite question: how small can the number of incidences be provided that points and tubes satisfy some natural spacing conditions? We show some results in this direction, in particular, we prove that if you choose n points in the unit square and draw a line through each point, then there exists a non-trivial point-line pair at distance at most n^{-2/3+o(1)}. This implies that among any n points in the unit square there are three spanning a triangle of area at most n^{-7/6+o(1)}, improving the previous upper bound for the Heilbronn's triangle problem.

Joint with Alex Cohen and Cosmin Pohoata.