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 $$Δ_y(E):={|x-y|:x\in E}$$ 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.