Hausdorff dimension of pinned distance sets and the 𝐿²-method

Liu, Bochen

doi:10.1090/proc/14740

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

Proc. Amer. Math. Soc. 148 (2020), 333-341

DOI: https://doi.org/10.1090/proc/14740

Published electronically: August 7, 2019

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Abstract:

We prove that for any compact set $E\subset \mathbb {R}^2$, $\dim _{\mathcal {H}}(E)>1$, there exists $x\in E$ such that the Hausdorff dimension of the pinned distance set \begin{equation*} \Delta _x(E)=\{|x-y|: y \in E\} \end{equation*} 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.

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