[Paper Review] Falconer distance problem, additive energy and Cartesian products

Alex Iosevich, Bochen Liu|arXiv (Cornell University)|Jun 25, 2015

Mathematical Dynamics and Fractals5 references3 citations

TL;DR

This paper improves the Falconer distance problem threshold for Cartesian products in $¹\mathbb{R}^2$ and higher dimensions by leveraging additive energy and Fourier analysis. It shows that if $E = A \times B$ with $\dim_H(A) + \dim_H(B) + \max(\dim_H(A), \dim_H(B)) > 2$, then $\Delta(E)$ has positive Lebesgue measure, improving the $\frac{4}{3}$ threshold in 2D and reducing Erdogan's $\frac{d}{2}+\frac{1}{3}$ to $\frac{d^2}{2d-1}$ in $d$ dimensions under Cartesian product structure.

ABSTRACT

A celebrated result due to Wolff says if $E$ is a compact subset of ${\Bbb R}^2$, then the Lebesgue measure of the distance set $Δ(E)=\{|x-y|: x,y \in E \}$ is positive if the Hausdorff dimension of $E$ is greater than $\frac{4}{3}$. In this paper we improve the $\frac{4}{3}$ barrier by a small exponent for Cartesian products. In higher dimensions, also in the context of Cartesian products, we reduce Erdogan's $\frac{d}{2}+\frac{1}{3}$ exponent to $\frac{d^2}{2d-1}$. The proof uses a combination of Fourier analysis and additive comibinatorics.

Motivation & Objective

• To improve the known threshold for the Falconer distance problem in $\mathbb{R}^d$ for sets that are Cartesian products of subsets of $\mathbb{R}$.
• To reduce the exponent required for positive Lebesgue measure of the distance set $\Delta(E)$ beyond the classical $\frac{d}{2}+\frac{1}{3}$ bound established by Erdogan.
• To investigate how the structure of Cartesian products, particularly through additive energy and Frostman measures, enables improved estimates in the Mattila integral.
• To extend Wolff's $\frac{4}{3}$ result in $\mathbb{R}^2$ to a broader class of product sets using Ahlfors-David regularity and improved additive energy bounds.

Proposed method

• Uses the Mattila integral as the primary tool to relate the decay of Fourier transforms of Frostman measures to the positivity of the distance set's Lebesgue measure.
• Applies the solid average estimate (Lemma 2.1) to bound $L^2$-restriction of Fourier transforms over spheres via measure growth conditions.
• Employs a parameterization of the sphere to decompose the Mattila integral into angular and radial components, focusing on the behavior near the equator.
• Leverages Dyatlov and Zahl's recent result on additive energy for Ahlfors-David regular sets to obtain improved decay estimates beyond the trivial $t^{-\alpha}$ bound.
• Combines the improved additive energy decay with the solid average estimate to refine the $L^2$-decay of the Fourier transform of the product measure.
• Uses the Mattila integral's convergence condition $\int |\hat{\mu}(\xi)|^2 |\xi|^{-\gamma} d\xi < \infty$ to derive the final threshold $s > \frac{d^2}{2d-1}$ in $d$ dimensions.

Experimental results

Research questions

• RQ1 Can the Falconer distance problem threshold be improved for Cartesian product sets in $\mathbb{R}^2$ beyond the $\frac{4}{3}$ bound?
• RQ2 Does Ahlfors-David regularity of the factors in a Cartesian product lead to improved decay in the Fourier transform of the associated Frostman measure?
• RQ3 Can the $\frac{d}{2}+\frac{1}{3}$ exponent in higher dimensions be reduced for product sets using additive energy and Fourier-analytic techniques?
• RQ4 What is the optimal dimension threshold for a Cartesian product $A_1 \times \cdots \times A_d$ to ensure the distance set has positive Lebesgue measure?
• RQ5 How does the imbalance in the dimensions of the factors affect the decay rate of the Fourier transform in the Mattila integral?

Key findings

• For $E = A \times B \subset \mathbb{R}^2$, if $\dim_H(A) + \dim_H(B) + \max(\dim_H(A), \dim_H(B)) > 2$, then $\Delta(E)$ has positive Lebesgue measure, improving the $\frac{4}{3}$ threshold.
• When $\dim_H(A) = \dim_H(B) = \alpha$ and $A$ is Ahlfors-David regular, there exists $\delta = \delta(C_{\nu_A}) > 0$ such that $\alpha > \frac{2}{3} - \delta$ implies $\Delta(E)$ has positive measure.
• In $\mathbb{R}^d$, for $E = A_1 \times \cdots \times A_d$, if $\sum_{j=1}^d s_j > \frac{d^2}{2d-1}$, then $\Delta(E)$ has positive Lebesgue measure, improving Erdogan's $\frac{d}{2} + \frac{1}{3}$ bound.
• The improved decay $t^{-\alpha - \delta}$ of the Fourier transform, derived from Dyatlov and Zahl's additive energy estimate, enables the threshold improvement in the Ahlfors-David regular case.
• The method achieves a quantitative improvement in the exponent by exploiting the imbalance in the Mattila integral and the structure of product sets.
• The final convergence condition for the Mattila integral is satisfied when $s > \frac{d^2}{2d-1}$, which is strictly less than $\frac{d}{2} + \frac{1}{3}$ for $d \geq 3$.