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Two-ends Furstenberg estimates in the plane

Hong Wang, Shukun Wu

TL;DR

This work addresses two-ends Furstenberg-type incidence bounds in the plane for Katz-Tao $(\delta,t)$-sets of lines, extending prior results to general $t\in[0,2]$. The authors develop a comprehensive framework combining shading, two-ends conditions, uniformization, branching functions, and multi-scale decomposition, and they integrate powerful Furstenberg-type results from recent literature to obtain a quantitative lower bound on the union of shadings. A key novelty is the appearance of the factor $\gamma_{Y,t^*}^{-1/2}$, which captures the non-uniform distribution of $Y(\ell)$ across scales and is necessary except in the symmetric case $t=1$. The results advance planar incidence geometry by enabling two-ends Furstenberg estimates for broad line families and illuminate the role of scale-sensitive distributional parameters in fractal incidence bounds.

Abstract

We prove two-ends Furstenberg estimates in the plane for a Katz-Tao $(δ,t)$-set of lines, for general $t\in[0,2]$.

Two-ends Furstenberg estimates in the plane

TL;DR

This work addresses two-ends Furstenberg-type incidence bounds in the plane for Katz-Tao -sets of lines, extending prior results to general . The authors develop a comprehensive framework combining shading, two-ends conditions, uniformization, branching functions, and multi-scale decomposition, and they integrate powerful Furstenberg-type results from recent literature to obtain a quantitative lower bound on the union of shadings. A key novelty is the appearance of the factor , which captures the non-uniform distribution of across scales and is necessary except in the symmetric case . The results advance planar incidence geometry by enabling two-ends Furstenberg estimates for broad line families and illuminate the role of scale-sensitive distributional parameters in fractal incidence bounds.

Abstract

We prove two-ends Furstenberg estimates in the plane for a Katz-Tao -set of lines, for general .

Paper Structure

This paper contains 3 sections, 22 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of the main theorem

Key Result

Theorem 1.3

Let $\delta\in(0,1)$. Let $(L,Y)_\delta$ be a set of directional $\delta$-separated lines in $\mathbb{R}^2$ with an $(\varepsilon_1, \varepsilon_2)$-two-ends shading such that $|Y(\ell)|\geq\lambda\delta$ for all $\ell\in L$. Then for all $\varepsilon>0$,

Theorems & Definitions (48)

  • Definition 1.1: Shading
  • Definition 1.2: Two-ends
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3: $(\delta,s,C)$-set
  • Definition 2.4: Katz-Tao $(\delta,s,C)$-set
  • ...and 38 more