Bochen Liu
Published 2018-10-18Version 1
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 $$\Delta_x(E)=\{|x-y|: y \in E\}$$ 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.
On the distance sets spanned by sets of dimension $d/2$ in $\mathbb{R}^d$
On the Hausdorff dimension of Furstenberg sets and orthogonal projections in the plane
Thickness and a gap lemma in $\mathbb{R}^d$
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