Lower bounds for incidences

Abstract

Let p 1 , … , p n be a set of points in the unit square and let T 1 , … , T n be a set of δ -tubes such that T j passes through p j . We 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, we show that in any configuration of points p 1 , … , p n ∈ [ 0 , 1 ] 2 along with a line ℓ j through each point p j , there exist j ≠ k for which d ( p j , ℓ k ) ≲ 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 our previous work arXiv: 2305.18253.