Lower bounds for incidences
Year: 2025
Type: article
Abstract: Let $p_{1},\ldots ,p_{n}$ p 1 , … , p n be a set of points in the unit square and let $T_{1},\ldots ,T_{n}$ T 1 , … , T n be a set of $\delta $ δ -tubes such that $T_{j}$ T j passes through $p_{j}$ 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 $p_{1},\ldots , p_{n} \in [0,1]^{2}$ p 1 , … , p n ∈ [ 0 , 1 ] 2 along with a line $\ell _{j}$ ℓ j through each point $p_{j}$ p j , there exist $j\neq k$ j ≠ k for which $d(p_{j}, \ell _{k}) \lesssim n^{-2/3+o(1)}$ d ( p j , ℓ k ) ≲ n − 2 / 3 + o ( 1 ) . It follows from the latter result that any set of $n$ n points in the unit square contains three points forming a triangle of area at most $n^{-7/6+o(1)}$ 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 .
Source: Inventiones mathematicae
Authors: Alex Cohen, Cosmin Pohoata, Dmitrii Zakharov
Institutions: Massachusetts Institute of Technology, Emory University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4408454108