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Incidence estimates for quasi-product sets and applications

Ciprian Demeter, William O'Regan

TL;DR

The paper develops Szemerédi–Trotter-type incidence bounds for δ-discretized Cartesian-product structures via Furstenberg-set techniques, introducing quasi-product and rectangular KT frameworks. It then applies these incidence bounds to Fourier-decay questions for fractal measures on curves (notably the parabola) and to discretised sum-product problems, obtaining dimension-lowering consequences for A+A and A/A and connecting to Elekes–Ruzsa and Solymosi strategies. A suite of auxiliary results on discretised sets and scale-compatibility underpins the incidence and Fourier analyses, while the high-low method, energy estimates for parabola, and decoupling provide complementary routes to Fourier decay. Together, these results advance understanding of fractal incidence geometry, Fourier-analytic behavior on curved manifolds, and fractal sum-product phenomena with concrete quantitative bounds across regimes of fractal dimension $s$.

Abstract

We use recent advances in the theory of Furstenberg sets to prove new incidence results of Szemerédi--Trotter strength for $δ$-discretized structures with Cartesian product flavor. We use these results to make progress on a number of problems that include energy estimates and Fourier decay of fractal measures supported on curves, as well as various sum-product-like results governed by fractal dimension.

Incidence estimates for quasi-product sets and applications

TL;DR

The paper develops Szemerédi–Trotter-type incidence bounds for δ-discretized Cartesian-product structures via Furstenberg-set techniques, introducing quasi-product and rectangular KT frameworks. It then applies these incidence bounds to Fourier-decay questions for fractal measures on curves (notably the parabola) and to discretised sum-product problems, obtaining dimension-lowering consequences for A+A and A/A and connecting to Elekes–Ruzsa and Solymosi strategies. A suite of auxiliary results on discretised sets and scale-compatibility underpins the incidence and Fourier analyses, while the high-low method, energy estimates for parabola, and decoupling provide complementary routes to Fourier decay. Together, these results advance understanding of fractal incidence geometry, Fourier-analytic behavior on curved manifolds, and fractal sum-product phenomena with concrete quantitative bounds across regimes of fractal dimension .

Abstract

We use recent advances in the theory of Furstenberg sets to prove new incidence results of Szemerédi--Trotter strength for -discretized structures with Cartesian product flavor. We use these results to make progress on a number of problems that include energy estimates and Fourier decay of fractal measures supported on curves, as well as various sum-product-like results governed by fractal dimension.

Paper Structure

This paper contains 20 sections, 42 theorems, 333 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Incidence estimates
  4. Fourier decay of fractal measures
  5. Sum-product problems
  6. Auxiliary results
  7. Properties of discretised sets
  8. Estimates for the number of $r$-heavy squares
  9. Fourier decay of fractal measures
  10. The high-low method
  11. Energy estimates for the parabola
  12. The decoupling method
  13. Sum-product problems
  14. Projection results
  15. Proof of Theorem \ref{['thm.dis']}
  16. ...and 5 more sections

Key Result

Theorem 1.3

Let $0<s,d<1$. Define Assume $\mathbb{T}$ is a $(\delta,s,d,K_1,K_2)$-quasi-product set. Consider a shading $Y$ of $\mathbb{T}$ such that $Y(T)$ is a $(\delta,\sigma,K_3)$-KT set for each $T\in\mathbb{T}$. Then we have

Theorems & Definitions (81)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • proof
  • Conjecture 1.7
  • Theorem 1.8
  • Theorem 1.9
  • ...and 71 more