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Incidence Estimates for Tubes in Complex Space

Sarah Tammen, Lingxian Zhang

TL;DR

The paper develops a complex-space incidence theory for δ-tubes, proving complex analogues of the Guth–Solomon–Wang bounds under strong spacing, and uses these results to obtain a discretized Falconer distance bound in $\mathbb{C}^2$ by translating difference sets into tube incidences via auxiliary lines. A two-scale, multiscale induction framework bridges geometric and Fourier analyses, enabling a dichotomy analysis (thin/thick) and a heavy-ball argument to bound the number of rich δ-balls. The results hinge on the construction of almost-$\delta$-caps, complex tube configurations, and a precise volume bound for intersections of δ-neighborhoods of complex lines, with subsections deriving quantitative auxiliary-line intersections and spacing properties. Together, these contribute a finite-field–style incidence theory in complex space and a new angle-based distance-set bound with potential applications to problems involving complex geometries.

Abstract

In this paper, we prove a complex version of the incidence estimate of Guth, Solomon and Wang for tubes obeying certain strong spacing conditions, and we use one of our new estimates to resolve a discretized variant of Falconer's distance set problem in $\mathbb{C}^2$.

Incidence Estimates for Tubes in Complex Space

TL;DR

The paper develops a complex-space incidence theory for δ-tubes, proving complex analogues of the Guth–Solomon–Wang bounds under strong spacing, and uses these results to obtain a discretized Falconer distance bound in by translating difference sets into tube incidences via auxiliary lines. A two-scale, multiscale induction framework bridges geometric and Fourier analyses, enabling a dichotomy analysis (thin/thick) and a heavy-ball argument to bound the number of rich δ-balls. The results hinge on the construction of almost--caps, complex tube configurations, and a precise volume bound for intersections of δ-neighborhoods of complex lines, with subsections deriving quantitative auxiliary-line intersections and spacing properties. Together, these contribute a finite-field–style incidence theory in complex space and a new angle-based distance-set bound with potential applications to problems involving complex geometries.

Abstract

In this paper, we prove a complex version of the incidence estimate of Guth, Solomon and Wang for tubes obeying certain strong spacing conditions, and we use one of our new estimates to resolve a discretized variant of Falconer's distance set problem in .
Paper Structure (8 sections, 14 theorems, 140 equations)

This paper contains 8 sections, 14 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Definitions, notations and conventions
  3. A bridge between different scales
  4. Proof of the main theorems
  5. Application to a variant of Falconer's distance set problem
  6. Proof of \ref{['prop:quantitative_auxline_intersection']}
  7. Proof of \ref{['prop:spacing_check']}
  8. Appendix

Key Result

Theorem 1.1

Let $\Theta$ be a partition of $\mathbb{C}\mathbb{P}^{n-1}$ into almost-$\delta$-caps. Suppose $1\leqslant W\leqslant\delta^{-1}$ and $1\leqslant N\leqslant W^{-1}\delta^{-1}$. For each $\theta\in\Theta$, let $\left\{T_{\theta,j}\right\}_{1\leqslant j\lesssim NW^{2(n-1)}}$ be a family of essentially

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Proposition 2.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • ...and 23 more