An $L^2$-identity and pinned distance problem

TL;DR

This paper links spherical average decay of Fourier transforms of measures to the pinned distance problem, improving known results by reducing the problem to integral identities involving group actions and Fourier analysis.

Contribution

It introduces a new identity connecting spherical averages with integral transforms, enabling improved bounds on the pinned distance problem using existing spherical average estimates.

Findings

Established a key identity relating spherical averages to Fourier transforms.

Reduced the pinned distance problem to an integral involving spherical averages.

Improved the dimensional threshold for the pinned distance problem using known estimates.

Abstract

Let $\mu$ be a Frostman measure on $E\subset\mathbb{R}^d$. The spherical average decay $$\int_{S^{d-1}}|\widehat{\mu}(r\omega)|^2\,d\omega\lesssim r^{-\beta} $$ was originally used to attack Falconer distance conjecture, via Mattila's integral. In this paper we consider the pinned distance problem, a stronger version of Falconer distance problem, and show that spherical average decay implies the same dimensional threshold on both of them. In particular, with the best known spherical average estimates, we improve Peres-Schlag's result on pinned distance problem significantly. The idea is to reduce the pinned distance problem to an integral where spherical averages apply. The key ingredient is the following identity. Using a group action argument, we show that for any Schwartz function $f$ on $\mathbb{R}^d$ and any $x\in\mathbb{R}^d$, $$\int_0^\infty |\omega_t*f(x)|^2\,t^{d-1}dt\,=\int_0^\infty|\widehat{\omega_r}*f(x)|^2\,r^{d-1}dr,$$ where $\omega_r$ is the normalized surface measure on $r S^{d-1}$. An interesting remark is that the right hand side can be easily seen equal to $$c_d\int\left|D_x^{-\frac{d-1}{2}}e^{-2\pi i t\sqrt{-\Delta}}f(x)\right|^2\,dt=c_d'\int\left|D_x^{-\frac{d-2}{2}}e^{2\pi i t\Delta}f(x)\right|^2\,dt.$$ An alternative derivation of Mattila's integral via group actions is also given in the Appendix.