Hausdorff dimension of pinned distance sets and the $L^2$-method

Abstract

We prove that for any $E\subset {\mathbb{R}}^2$, ${\dim }_{\mathcal{H}}(E)>1$, there exists $x\in E$ such that the Hausdorff dimension of the pinned distance set $${\Delta }_x(E)=\{|x-y|:y\in E\}$$ is no less than $\min \left\{\frac{4}{3}{\dim }_{\mathcal{H}}(E)-\frac{2}{3},1\right\}$. 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 $\epsilon >0$ there exists $x\in E$ such that the Hausdorff dimension of ${\Delta }_x(E)$ is at least $\min \left\{\frac{4}{3}{\dim }_{\mathcal{H}}(E)-\frac{2}{3}-\epsilon ,1\right\}$, and it implies the Hausdorff dimension of the distance set, $\Delta (E)=\{|x-y|:x,y\in E\}$, is at least $\min \left\{\frac{4}{3}{\dim }_{\mathcal{H}}(E)-\frac{2}{3},1\right\}$. 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.)