Lower bounds for incidences

TL;DR

This paper establishes lower bounds on incidences between points and tubes in the unit square, leading to new bounds on triangle areas and advancing understanding in geometric combinatorics.

Contribution

It provides the first lower bounds for point-tube incidences under regularity conditions and derives improved bounds for Heilbronn's triangle problem.

Findings

Lower bounds for incidences between points and tubes under regularity conditions.

Existence of point pairs with small distance to each other's lines.

New upper bounds for the smallest triangle area among points.

Abstract

Let $p_1,\ldots,p_n$ be a set of points in the unit square and let $T_1,\ldots,T_n$ be a set of $\delta$-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 $p_1,\ldots, p_n \in [0,1]^2$ along with a line $\ell_j$ through each point $p_j$, there exist $j\neq k$ for which $d(p_j, \ell_k) \lesssim n^{-2/3+o(1)}$. 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.