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

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.

23 pages. V3: A proof of Lemma 3.1 is added for completeness, and a typo on ${\omega }_Q(T)$ in Proposition 1.2 is corrected. V2: Theorem 1.2 is relabelled to Proposition 1.2, and the technique is now named "L2 ball inflation"