Cohen 2024 lower bounds incidences

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corollary_1_2: Given n >= 2 points of the unit square and a line through each, some point lies within distance C(eps) n^(-2/3+eps) of the line through a different point.

corollary_1_3: For points of the unit square with a delta-tube through each, the number of point-tube incidences is at least c(eps) delta^(3/2+eps) |P||T|.

theorem_1_1: For every eps > 0 and delta < delta_0(eps), any n >= delta^(-3/2-eps) points of the unit square, each with a delta-tube through it, have a point lying in the tube of another point.

theorem_1_4: For t in [1,2] and s in [0,1] with 2t+s > 3 there is eta(t,s) > 0 such that, for delta < delta_0(t,s), delta^(-t) points of the unit square, each carrying a (delta,s,delta^(-eta))-set of delta-tubes through it, have a point lying in a tube of another point.

theorem_1_8: For every eps > 0, every set of n points in the unit square contains a triangle of area at most C(eps) n^(-7/6+eps), so Heilbronn's triangle quantity for the square is at most n^(-7/6+o(1)).

theorem_1_9: For alpha, beta in [1,2] with alpha+beta > 3 and eps > 0 there is eta(alpha,beta,eps) > 0 such that every (delta,alpha,beta,delta^(-eta))-set X of point-line pairs, for delta < delta_0(alpha,beta,eps), has smoothed incidence count at least delta^(1+eps)|X|^2.

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Alex Cohen, Cosmin Pohoata, Dmitrii Zakharov, Lower bounds for incidences. Invent. Math. 240 (2025), no. 3, 1045-1118. DOI 10.1007/s00222-025-01331-2. arXiv:2409.07658. The copy read for this card is the arXiv version arXiv:2409.07658v2 (18 March 2025); labels below follow it.

The paper proves lower bounds for incidences between n points in the unit square and delta-tubes constrained so that tube T_j passes through point p_j. Theorem 1.1 states that for every eps > 0 and delta < delta_0(eps), if n >= delta^{-3/2-eps} then some nontrivial incidence p_j in T_k with j != k exists; the equivalent Corollary 1.2 says that given a line l_j through each p_j there are j != k with d(p_j, l_k) <= C(eps) n^{-2/3+eps}, and Corollary 1.3 gives I(P,T) >= c(eps) delta^{3/2+eps}|P||T| by subsampling. Theorem 1.4 generalizes to a whole (delta,s,delta^{-eta})-family of tubes through each of delta^{-t} points under 2t+s>3, for some eta(t,s) > 0, and Theorem 1.5 recovers a weaker form of an incidence lower bound of Dabrowski, Goering and Orponen for t+s>2. The framework uses Frostman-type regularity of point and line sets, which bounds how much they concentrate in w-balls at every scale w in [delta,1] and so gives lower bounds on their covering numbers at every scale. The paper's main result, Theorem 1.9, is a lower bound I(delta; X) >= delta^{1+eps}|X|^2 for the smoothed incidence count of a set X of point-line pairs satisfying a Frostman condition over phase-space rectangles of all side ratios with exponents alpha + beta > 3; Theorem 1.4 is deduced from it. The stated consequence (Theorem 1.8) is that any n points in the unit square contain a triangle of area at most n^{-7/6+o(1)}, which improves the authors' earlier n^{-8/7-1/2000} and, in the abstract's words, "attains the high-low limit established in our previous work".

Source: https://arxiv.org/abs/2409.07658. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2409.07658), every other right reserved.

Bears on. #507: Theorem 1.8 bounds the smallest triangle among n points of the unit square by n^{-7/6+o(1)}; the problem asks for the order of the analogous quantity for the unit disk, and the transfer to the disk and the resulting upper bound are recorded on the problem's claim page for this paper. No result here gives a lower bound.

Results. Labels and pages are those of arXiv:2409.07658v2. Read depth: claims checked for each statement below; the proofs were read for structure only.

• Theorem 1.1 (p. 1): for every eps > 0 and delta < delta_0(eps), n >= delta^{-3/2-eps} points of [0,1]^2 with a delta-tube T_j through each p_j have a nontrivial incidence p_j in T_k, j != k.
• Corollary 1.2 (p. 1): for an integer n >= 2 and n points of [0,1]^2 with a line l_j through each p_j, there are j != k with d(p_j, l_k) <~_eps n^{-2/3+eps}; stated to be equivalent to Theorem 1.1.
• Corollary 1.3 (p. 2): in the setting of Theorem 1.1, I(P,T) >~_eps delta^{3/2+eps}|P||T|, by subsampling.
• Theorem 1.4 (p. 2): for t in [1,2], s in [0,1] with 2t+s > 3 there is eta(t,s) > 0 such that, for delta < delta_0(t,s), any delta^{-t} points of [0,1]^2, each carrying a (delta,s,delta^{-eta})-set of delta-tubes through it, have a point in a tube of another point.
• Theorem 1.9 (p. 6): for alpha, beta in [1,2] with alpha + beta > 3 and eps > 0 there is eta(alpha,beta,eps) > 0 such that, for delta < delta_0(alpha,beta,eps), every (delta,alpha,beta,delta^{-eta})-set X in phase space has I(delta; X) >= delta^{1+eps}|X|^2.
• Theorem 1.8 (p. 4): for every eps > 0, every n points in the unit square contain a triangle of area <~_eps n^{-7/6+eps}.

Theorem 1.5 (p. 2), the weaker form of the Dabrowski-Goering-Orponen bound, and Proposition B.1 (Appendix B), a conditional route to Heilbronn's problem for k-gons, have no page here.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.