Lower bounds for incidences
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222_2025_Article_1331.pdf
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1.7 MB
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Author(s)
Cohen, Alex
Pohoata, Cosmin
Zakharov, Dmitrii
Date Issued
March 14, 2025
Journal
Inventiones mathematicae
Publisher
Springer Berlin Heidelberg
Citation
Cohen, A., Pohoata, C. & Zakharov, D. Lower bounds for incidences. Invent. math. 240, 1045–1118 (2025).
Version
Final published version
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution
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DOI of Published Version
https://doi.org/10.1007/s00222-025-01331-2
