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arXiv · 2603.15328

Lebesgue measure of distance sets with regular pins and multi-scale Mizohata-Takeuchi-type estimates

Bochen Liu
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Abstract

Suppose $E, F$ are Borel sets in the plane, $\dim_{\mathcal{H}} E>1$, $\dim_{\mathcal{H}} E+\dim_{\mathcal{H}} F>2$, and $F$ has equal Hausdorff and packing dimension. We prove that there exists $y\in F$ such that the pinned distance set $$\Delta_y(E):=\{|x-y|:x\in E\}$$ has positive Lebesgue measure. In particular, it settles the regular case of the distance set problem in the plane. The main ingredients of the proof consist of a multi-scale Good-Bad decomposition and a multi-scale Mizohata-Takeuchi-type estimate with arbitrary small power-loss.

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Bochen Liu. 2026-03-16. Lebesgue measure of distance sets with regular pins and multi-scale Mizohata-Takeuchi-type estimates. https://arxiv.org/abs/2603.15328

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