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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Hausdorff dimension of pinned distance sets and the $L^2$-method
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by Bochen Liu PDF
Proc. Amer. Math. Soc. 148 (2020), 333-341 Request permission


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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Additional Information
  • Bochen Liu
  • Affiliation: Department of Mathematics, the Chinese University of Hong Kong, Shatin, N.T., Hong Kong
  • MR Author ID: 1066951
  • Email:
  • Received by editor(s): May 13, 2019
  • Published electronically: August 7, 2019
  • Additional Notes: The author was supported by the grant CUHK24300915 from the Hong Kong Research Grant Council
  • Communicated by: Alexander Iosevich
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 148 (2020), 333-341
  • MSC (2010): Primary 28A75; Secondary 42B20
  • DOI:
  • MathSciNet review: 4042855