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Lower bounds for incidences

Alex Cohen, Cosmin Pohoata, Dmitrii Zakharov

TL;DR

This work establishes lower bounds for incidences between a point set and a corresponding family of $\delta$-tubes in the unit square under Frostman-type regularity, and shows strong Heilbronn-type consequences. The authors develop a phase-space framework for point-line pairs, extract uniform subsets, and apply a two-step incidence strategy combining an initial estimate with a high-low induction guided by Lipschitz branching functions. A central outcome is the nontrivial bound $I(\delta; \mathbf X) \gtrsim \delta^{1+\varepsilon} |\oldsymbol{\\mathbf X}|^2$ and the existence of an extra incidence, which translate into the distance bound $d(p_j, \ell_k) \lesssim n^{-2/3+o(1)}$ and Heilbronn triangle bound $\Delta \lesssim n^{-7/6+o(1)}$. The paper also connects to finite-field analogues, Furstenberg set estimates, and Kahane-type multiscale analysis, offering a new route to Heilbronn-type problems via a robust incidence-lowering framework. Overall, the results advance incidence geometry by delivering sharp lower bounds under broad regularity assumptions and by integrating a phase-space, Lipschitz-function approach with a multiscale high-low method. The techniques have potential applications to higher-dimensional incidence problems and to related extremal combinatorial geometry questions.

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 $δ$-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 $p_1,\ldots, p_n \in [0,1]^2$ 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.

Lower bounds for incidences

TL;DR

This work establishes lower bounds for incidences between a point set and a corresponding family of -tubes in the unit square under Frostman-type regularity, and shows strong Heilbronn-type consequences. The authors develop a phase-space framework for point-line pairs, extract uniform subsets, and apply a two-step incidence strategy combining an initial estimate with a high-low induction guided by Lipschitz branching functions. A central outcome is the nontrivial bound and the existence of an extra incidence, which translate into the distance bound and Heilbronn triangle bound . The paper also connects to finite-field analogues, Furstenberg set estimates, and Kahane-type multiscale analysis, offering a new route to Heilbronn-type problems via a robust incidence-lowering framework. Overall, the results advance incidence geometry by delivering sharp lower bounds under broad regularity assumptions and by integrating a phase-space, Lipschitz-function approach with a multiscale high-low method. The techniques have potential applications to higher-dimensional incidence problems and to related extremal combinatorial geometry questions.

Abstract

Let be a set of points in the unit square and let be a set of -tubes such that passes through . 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 along with a line through each point , there exist for which . It follows from the latter result that any set of points in the unit square contains three points forming a triangle of area at most . This new upper bound for Heilbronn's triangle problem attains the high-low limit established in our previous work arXiv:2305.18253.
Paper Structure (44 sections, 40 theorems, 355 equations, 1 figure)

This paper contains 44 sections, 40 theorems, 355 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Background on Heilbronn's triangle problem
  4. Application to Heilbronn's triangle problem
  5. Frostman regularity over rectangles and the main incidence lower bound
  6. Finite fields
  7. Prior work
  8. Paper outline
  9. Notation
  10. Proof sketch
  11. Setup
  12. Two-step strategy for finding incidences
  13. Examples of $(\delta, \alpha, \beta, C)$-sets
  14. The analysis of Lipschitz functions
  15. The space of branching functions
  16. ...and 29 more sections

Key Result

Theorem 1.1

For all $\varepsilon > 0$ the following holds for $\delta < \delta_0(\varepsilon)$. Let $p_1,\ldots, p_n$ be a set of points in $[0,1]^2$ along with a $\delta$-tube $T_j$ through each point. If $n \geqslant \delta^{-3/2-\varepsilon}$, there is some nontrivial incidence $p_j \in T_k$ where $j \neq k$

Figures (1)

  • Figure 1: If we rescale a $u\times uw$ rectangle to the unit square, a $\delta$-tube pointing in the long direction of the rectangle maps to a $\delta/uw$-tube.

Theorems & Definitions (86)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6: CPZ
  • Theorem 1.7: High-low inequality
  • Theorem 1.8
  • proof
  • Theorem 1.9
  • ...and 76 more