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Tangency counting for well-spaced circles

Dominique Maldague, Alexander Ortiz

TL;DR

The paper addresses a discrete circle tangency counting problem by exploiting a continuum-incidence viewpoint via incidences between points in $\mathbb{R}^3$ and light rays. It introduces a lifting to lightplanks and a novel stopping-time argument together with a refined decoupling theorem for the light cone in $\mathbb{R}^3$, culminating in sharp bounds for the number of $\mu$-rich tangency rectangles and, consequently, for tangency sites of well-spaced circle collections. The main achievement is breaking the $N^{3/2}$ barrier for well-spaced circles, yielding a bound of $|X|^{25/18+\varepsilon}$ on tangency sites, up to $\varepsilon$-loss, and establishing the sharpness of the method through probabilistic constructions. The results advance continuum incidence geometry techniques and provide a framework—via square-function refined decoupling—that could extend to other geometric incidence problems and Kakeya-type questions.

Abstract

In the late 90's, Tom Wolff introduced the circle tangency counting problem in his expository article on the Kakeya conjecture. For collections of well-spaced circles, we break the $N^{3/2}$-barrier, proving that a set of $N$ well-spaced circles has at most $N^{25/18+\varepsilon}$ sites of internal tangency. The circle tangency problem can be related to a problem about incidences between points in $\mathbb{R}^3$ and light rays. For this problem, we introduce a stopping time argument to extract maximal information about well-spaced points from a refined decoupling theorem for the light cone in $\mathbb{R}^3$, leading to sharp bounds on the number of $μ$-rich tangency rectangles.

Tangency counting for well-spaced circles

TL;DR

The paper addresses a discrete circle tangency counting problem by exploiting a continuum-incidence viewpoint via incidences between points in and light rays. It introduces a lifting to lightplanks and a novel stopping-time argument together with a refined decoupling theorem for the light cone in , culminating in sharp bounds for the number of -rich tangency rectangles and, consequently, for tangency sites of well-spaced circle collections. The main achievement is breaking the barrier for well-spaced circles, yielding a bound of on tangency sites, up to -loss, and establishing the sharpness of the method through probabilistic constructions. The results advance continuum incidence geometry techniques and provide a framework—via square-function refined decoupling—that could extend to other geometric incidence problems and Kakeya-type questions.

Abstract

In the late 90's, Tom Wolff introduced the circle tangency counting problem in his expository article on the Kakeya conjecture. For collections of well-spaced circles, we break the -barrier, proving that a set of well-spaced circles has at most sites of internal tangency. The circle tangency problem can be related to a problem about incidences between points in and light rays. For this problem, we introduce a stopping time argument to extract maximal information about well-spaced points from a refined decoupling theorem for the light cone in , leading to sharp bounds on the number of -rich tangency rectangles.

Paper Structure

This paper contains 14 sections, 11 theorems, 152 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Circle tangencies and unit distances
  3. Discrete versus continuum problems
  4. Decoupling and the proof of Theorem \ref{['thm:rect1']}
  5. Outline of the paper
  6. Notation
  7. Acknowledgments
  8. Estimate of mu-rich tangency rectangles
  9. Decoupling
  10. Proof of the refined decoupling Theorem \ref{['Grefdec']}
  11. Refined decoupling for the parabola (Proof of Theorem \ref{['Grefdec']} part 1/2)
  12. Refined decoupling for the cone (Proof of Theorem \ref{['Grefdec']} part 2/2)
  13. Sharpness
  14. Large deviations for sums of Bernoulli random variables

Key Result

Theorem 1.1

For every $\varepsilon>0$, there is a constant $A_\varepsilon$ so that the following holds. Let $X\subset [0,1]^2\times[1,2]$ be well-spaced, and let $\mathcal{C}_X$ denote the collection of circles where $C_{z,r}$ denotes the circle with center $z\in\mathbb{R}^2$ and radius $r>0$. If no three circles of $\mathcal{C}_X$ are tangent at a point, then the following estimate holds:

Figures (3)

  • Figure 1: A clamshell of $5$ circles.
  • Figure 2: A lattice $X$ (points above) and a selected subset of circles $\mathcal{C}_X$ (below), with black dots marking the sites of internal tangencies between circles.
  • Figure 3: A set of $6$ circles with $|\mathcal{T}(\mathcal{C})| = 6$.

Theorems & Definitions (22)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Theorem 2.1
  • proof
  • Proposition 2.2: Locally constant property
  • proof : Proof of Proposition \ref{['prop:loc-const']}; the locally constant property
  • Remark 2.1
  • Theorem 2.3
  • ...and 12 more