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

Bochen Liu

Published 2018-02-01Version 1

We prove that if the Hausdorff dimension of $E\subset\mathbb{R}^d$, $d\geq 2$ is greater than $\frac{d}{2}+\frac{1}{3}$, there exists $x\in E$ such that the pinned distance set $$\Delta_x(E)=\{|y-x|: y\in E \}$$ has positive Lebesgue measure. This improves a result of Peres and Schlag. The key new ingredient in our proof 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 equals $$c_d\int_0^\infty\left|D_x^{-\frac{d-1}{2}}e^{-2\pi i t\sqrt{-\Delta}}f(x)\right|^2\,dt=c_d'\int_0^\infty\left|D_x^{-\frac{d-2}{2}}e^{2\pi i t\Delta}f(x)\right|^2\,dt,$$ where $\Delta$ is the standard Laplacian and $D_x^\alpha=(-\Delta)^{\frac{\alpha}{2}}$.