Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Incidence bounds related to circular Furstenberg sets

John Green, Terence L. J. Harris, Yumeng Ou, Kevin Ren, Sarah Tammen

TL;DR

This work develops incidence-based bounds between families of circles, sine waves, and cinematic curves and uses these to derive dimension lower bounds for circular and sine-wave Furstenberg sets in the plane. By combining high–low frequency decompositions with local smoothing (and its variable-coefficient generalization) and trilinear Fourier restriction, the authors obtain new curved Furstenberg bounds that improve upon prior results and apply to cinematic curves as well. A key contribution is the systematic transfer from incidence bounds to Furstenberg-dimension estimates, supported by a sharp projection-type sharpness discussion. They also establish positive-area results for sets containing many cinematic curves and provide sharpness examples for the corresponding projection bounds.

Abstract

We prove bounds on approximate incidences between families of circles and families of points in the plane. As a consequence, we prove a lower bound for the dimension of circular $(u,v)$-Furstenberg sets, which is new for large $u$ and $v$.

Incidence bounds related to circular Furstenberg sets

TL;DR

This work develops incidence-based bounds between families of circles, sine waves, and cinematic curves and uses these to derive dimension lower bounds for circular and sine-wave Furstenberg sets in the plane. By combining high–low frequency decompositions with local smoothing (and its variable-coefficient generalization) and trilinear Fourier restriction, the authors obtain new curved Furstenberg bounds that improve upon prior results and apply to cinematic curves as well. A key contribution is the systematic transfer from incidence bounds to Furstenberg-dimension estimates, supported by a sharp projection-type sharpness discussion. They also establish positive-area results for sets containing many cinematic curves and provide sharpness examples for the corresponding projection bounds.

Abstract

We prove bounds on approximate incidences between families of circles and families of points in the plane. As a consequence, we prove a lower bound for the dimension of circular -Furstenberg sets, which is new for large and .

Paper Structure

This paper contains 18 sections, 17 theorems, 278 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Description of the approach
  3. Connection to restricted families of projections
  4. Sets containing many cinematic curves
  5. Notation
  6. Outline
  7. Acknowledgments
  8. Incidence bounds
  9. Circle incidences via local smoothing
  10. Broad incidences between circles and discs
  11. Broad incidences between sine waves and discs
  12. Cinematic curve incidences via variable coefficient local smoothing
  13. Converting incidence bounds to lower bounds for curved Furstenberg sets
  14. Positive area of sets containing many cinematic curves
  15. Sharpness of Proposition \ref{['projection']} when s=t/3
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $0 < u \leq 1$ and $0 \leq v \leq 3$. If $F \subseteq \mathbb{R}^2$ is a circular $(u,v)$-Furstenberg set, or a sine wave $(u,v)$-Furstenberg set, then and and (by liu in the circular case)

Figures (1)

  • Figure 1: The current best known lower bounds for circular Furstenberg sets, including our results in Theorem \ref{['curvedfurstenberg']}. The known bounds for sine curves are the same, but the lower bound $\min\{u+v,2u\}$ is from zahl in this case, and $u+\frac{v}{3}$ is Theorem \ref{['curvedfurstenberg']}.

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 1.3
  • Conjecture 1.4
  • Conjecture 1.5
  • Theorem 1.6
  • Theorem 2.1
  • proof
  • Definition 2.2
  • Theorem 2.3
  • ...and 26 more