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Szemerédi-Trotter bounds for tubes and applications

Ciprian Demeter, Hong Wang

TL;DR

This work establishes sharp Szemerédi–Trotter type bounds for incidences between planar tubes under packing constraints and translates these bounds into Fourier-analytic estimates for fractal measures supported on planar curves. The authors blend uniform Katz–Tao sets, probabilistic uniformization, Furstenberg-set techniques, and multi-scale decompositions (including two-ends and high–low analyses) to derive a robust tube-incidence bound: for $s\in(0,\tfrac12]$ and appropriate $\delta$, $|\mathcal{P}_r(\mathcal{T})| \lesssim_\upsilon \delta^{-\upsilon}\frac{|\mathcal{T}|^2}{r^3}$ for $1\le r\lesssim \delta^{-s}$. Consequences include refined $L^6$ bounds on the Fourier transform of fractal measures supported on planar curves, improving exponents in the fractal restriction context. The approach advances incidence geometry with packing assumptions and leverages Furstenberg-type bounds to connect geometric configuration to harmonic-analytic estimates, yielding both sharp incidence results and applications to fractal Fourier decay. The methods—probabilistic uniformization, two-ends reductions, and induction on scales—offer a flexible framework for related geometric-probination problems and their Fourier-analytic consequences.

Abstract

We prove sharp estimates for incidences involving planar tubes that satisfy packing conditions. We apply them to improve the estimates for the Fourier transform of fractal measures supported on planar curves.

Szemerédi-Trotter bounds for tubes and applications

TL;DR

This work establishes sharp Szemerédi–Trotter type bounds for incidences between planar tubes under packing constraints and translates these bounds into Fourier-analytic estimates for fractal measures supported on planar curves. The authors blend uniform Katz–Tao sets, probabilistic uniformization, Furstenberg-set techniques, and multi-scale decompositions (including two-ends and high–low analyses) to derive a robust tube-incidence bound: for and appropriate , for . Consequences include refined bounds on the Fourier transform of fractal measures supported on planar curves, improving exponents in the fractal restriction context. The approach advances incidence geometry with packing assumptions and leverages Furstenberg-type bounds to connect geometric configuration to harmonic-analytic estimates, yielding both sharp incidence results and applications to fractal Fourier decay. The methods—probabilistic uniformization, two-ends reductions, and induction on scales—offer a flexible framework for related geometric-probination problems and their Fourier-analytic consequences.

Abstract

We prove sharp estimates for incidences involving planar tubes that satisfy packing conditions. We apply them to improve the estimates for the Fourier transform of fractal measures supported on planar curves.
Paper Structure (9 sections, 27 theorems, 264 equations, 1 figure)

This paper contains 9 sections, 27 theorems, 264 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Probabilistic arguments
  4. Consequences of Theorem \ref{['thm: main']}
  5. Incidence results via Furstenberg set estimates
  6. The high-low method
  7. Two-ends reduction
  8. Proof of Theorem \ref{['thm: main']}
  9. Proof of Theorem \ref{['t1']}

Key Result

Theorem 1.1

Let $s\in (0, 1/2]$. Suppose $\Lambda\subset \varmathbb{S}^1$ is a $(\delta, s)$-set consisting of $\delta$-intervals (arcs) with cardinality $|\Lambda|\sim \delta^{-s}$. Suppose that for each $\delta$-arc $\theta\in \Lambda$ there is a $(\delta, 1-s)$-set $\mathcal{T}_{\theta}$ of $\delta$-tubes in

Figures (1)

  • Figure :

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5: RW
  • Corollary 1.6
  • proof
  • Definition 2.1: $(\delta, s,C)$-set
  • Definition 2.2: $(\delta,s, K)$-Katz-Tao set
  • Definition 2.3: Uniform sets
  • ...and 46 more