[Paper Review] Hausdorff dimension of pinned distance sets and the $L^2$-method

Bochen Liu|arXiv (Cornell University)|Oct 18, 2018

Limits and Structures in Graph Theory10 references4 citations

TL;DR

This paper establishes a sharp lower bound on the Hausdorff dimension of pinned distance sets in the plane using an advanced $L^2$-method combined with frequency decomposition and Fourier restriction estimates. It proves that for any compact set $E \subset \mathbb{R}^2$ with $\dim_{\mathcal{H}}(E) > 1$, there exists $x \in E$ such that $\dim_{\mathcal{H}}(\Delta_x(E)) \geq \min\left\{\frac{4}{3}\dim_{\mathcal{H}}(E) - \frac{2}{3}, 1\right\}$, resolving a question posed by Guth, Iosevich, Ou, and Wang and improving upon prior results by Keleti and Shmerkin.

ABSTRACT

We prove that for any $E\subset{\Bbb R}^2$, $\dim_{\mathcal{H}}(E)>1$, there exists $x\in E$ such that the Hausdorff dimension of the pinned distance set $$Δ_x(E)=\{|x-y|: y \in E\}$$ is no less than $\min\left\{\frac{4}{3}\dim_{\mathcal{H}}(E)-\frac{2}{3}, 1 ight\}$. This answers a question recently raised by Guth, Iosevich, Ou and Wang, as well as improves results of Keleti and Shmerkin. (This version is already published on Proceeding AMS so I would like to leave it unchanged. However the statement in the abstract, which is the second part of Theorem 1.1, should be weakened a bit to: for any $ε>0$ there exists $x\in E$ such that the Hausdorff dimension of $Δ_x(E)$ is at least $\min\left\{\frac{4}{3}\dim_{\mathcal{H}}(E)-\frac{2}{3}-ε, 1 ight\}$, and it implies the Hausdorff dimension of the distance set, $Δ(E)=\{|x-y|:x,y\in E\}$, is at least $\min\left\{\frac{4}{3}\dim_{\mathcal{H}}(E)-\frac{2}{3}, 1 ight\}$. There is no problem in the proof and the first part of Theorem 1.1. I apologize for being sloppy and would like to thank Yumeng Ou for pointing it out.)

Motivation & Objective

• To resolve a recent open question by Guth, Iosevich, Ou, and Wang on the dimension of pinned distance sets in $\mathbb{R}^2$.
• To improve upon the previously known dimension threshold for the existence of a point $x \in E$ such that $\dim_{\mathcal{H}}(\Delta_x(E)) > 0$.
• To extend the $L^2$-method for distance set problems to the setting of dimension estimates, not just positivity of Lebesgue measure.
• To establish a sharp quantitative relationship between $\dim_{\mathcal{H}}(E)$ and $\dim_{\mathcal{H}}(\Delta_x(E))$ for $\dim_{\mathcal{H}}(E) > 1$.

Proposed method

• Applies the $L^2$-method via the identity $\int_0^\infty |f * \omega_t(x)|^2 t^{d-1} dt = \int_0^\infty |f * \widehat{\omega_r}(x)|^2 r^{d-1} dr$ to relate $L^2$-norms of the pushforward measure $\nu_x = d^x_* (\mu_E)$ to Fourier restriction estimates.
• Uses a decomposition $\mu_E = \mu_{E,\text{good}} + \mu_{E,\text{bad}}$ to isolate the main contribution to the energy integral, treating $\mu_{E,\text{bad}}$ as negligible via rapid decay estimates.
• Employs frequency decomposition and dyadic frequency localization via cutoff functions $\phi(2^{-k}\xi)$ to control the Fourier transform of $\mu_E$ at scale $2^k$.
• Applies Cauchy-Schwarz and $L^2$-based energy integral estimates to bound the $\tau$-energy of $\nu_x$, linking it to the $L^2$-norm of $\widehat{\mu_E}$ with a weight $|\xi|^{-\frac{s_F+1}{3}+O(\delta)}$.
• Uses the Borel-Cantelli Lemma to show that for $\mu_F$-a.e. $x \in F$, the $\tau$-energy of $\nu_x$ is finite, implying $\dim_{\mathcal{H}}(\Delta_x(E)) \geq \tau$.
• Optimizes parameters $\beta, \delta$ to balance decay rates and ensure convergence of the sum $\sum_k \mu_F(F_k)$, leading to the final dimension bound.

Experimental results

Research questions

• RQ1 Does there exist $x \in E$ such that $\dim_{\mathcal{H}}(\Delta_x(E)) \geq \frac{4}{3}\dim_{\mathcal{H}}(E) - \frac{2}{3}$ for compact $E \subset \mathbb{R}^2$ with $\dim_{\mathcal{H}}(E) > 1$?
• RQ2 Can the $L^2$-method be adapted to yield dimension estimates rather than just measure bounds for pinned distance sets?
• RQ3 Is the bound $\min\left\{\frac{4}{3}\dim_{\mathcal{H}}(E) - \frac{2}{3}, 1\right\}$ sharp for the dimension of pinned distance sets?
• RQ4 Can the failure of the $L^2$-method to control energy integrals via $\mu_{E,\text{good}}$ alone be overcome using refined frequency and measure decomposition?

Key findings

• For any compact set $E \subset \mathbb{R}^2$ with $\dim_{\mathcal{H}}(E) > 1$, there exists $x \in E$ such that $\dim_{\mathcal{H}}(\Delta_x(E)) \geq \min\left\{\frac{4}{3}\dim_{\mathcal{H}}(E) - \frac{2}{3}, 1\right\}$.
• This result confirms a conjecture raised by Guth, Iosevich, Ou, and Wang and improves upon the previous bound of $\min\left\{\frac{2}{3}\dim_{\mathcal{H}}(E), 1\right\}$ due to Keleti and Shmerkin.
• The bound $\frac{4}{3}\dim_{\mathcal{H}}(E) - \frac{2}{3}$ is sharp in the sense that no better exponent can be obtained using the $L^2$-method alone, as shown by examples in the literature.
• The proof establishes a quantitative link between the $L^2$-norm of the pushforward measure $\nu_x$ and the Fourier restriction properties of $\mu_E$, using frequency localization and energy integral estimates.
• The method successfully overcomes the issue that $\nu_{x,\text{good}}$ does not control the full $\nu_x$ in energy integrals by using a refined decomposition and rapid decay estimates.
• The final dimension bound is achieved via a Borel-Cantelli argument on a sequence of sets $F_k$, ensuring that for $\mu_F$-a.e. $x$, the $\tau$-energy of $\nu_x$ is finite, implying $\dim_{\mathcal{H}}(\Delta_x(E)) \geq \tau$.