Alex Cohen is a Ph.D. Student at MIT, advised by Dr. Larry Guth. Alex is expected to graduate in Spring 2025 and is going to be an assistant professor at Courant Institute of Mathematical Sciences, NYU, starting September 2025.
In this talk, Alex presents his joint work with Cosmin Pohoata and Dmitrii Zakharov (https://arxiv.org/abs/2409.07658 ). They prove a lower bound for the number of incidences between the points and tubes under a natural regularity condition (similar to Frostman regularity). As a consequence, they show that in any configuration of points p_1,…,p_n∈[0,1]^2 along with a line ℓj through each point p_j, there exist j≠k for which d(pj,ℓk)≲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 Heilbronn’s triangle problem attains the high-low limit established in their previous work arXiv:2305.18253.
Keywords: incidences, fractal geometry, Heilbronn triangle problem.
MSC2020: 52C99.