Improvement on 2-Chains Inside Thin Subsets of Euclidean Spaces

  • Bochen LiuEmail author


Given \(E\subset \mathbb {R}^d\), \(d\ge 2\), we prove that when \(\dim _{{\mathcal {H}}}(E)>\frac{d}{2}+\frac{1}{3}\), the set of gaps of 2-chains inside E, i.e.,
$$\begin{aligned} \Delta _2(E)=\{(|x-y|, |y-z|): x, y, z\in E \}\subset \mathbb {R}^2 \end{aligned}$$
has positive Lebesgue measure. This improves a result of Bennett, Iosevich, and Taylor. We also consider when the set of similarity classes of 2-chains,
$$\begin{aligned} S_2(E)=\left\{ \frac{t_1}{t_2}:(t_1,t_2)\in \Delta _2(E)\right\} =\left\{ \frac{|x-y|}{|y-z|}: x, y, z\in E \right\} \subset \mathbb {R} \end{aligned}$$
has positive Lebesgue measure. The main idea in this paper is to reduce geometric problems to integrals where Wolff-Erdoğan’s spherical averaging estimates apply. Invariant measures on orthogonal groups play an important role in the reduction.


Distance problem Spherical averages Falconer distance conjecture Spherical averages Group action reduction Chains 

Mathematics Subject Classification

28A75 42B20 



This work was done when the author was visiting the Hong Kong University of Science and Technology. The author would like to thank Yang Wang for the financial support.


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Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.Department of MathematicsBar-Ilan UniversityRamat GanIsrael

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