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

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 $${\mathrm{\Delta }}_y(E):=\{\mathrm{\mid }x-y\mathrm{\mid }:x\in E\}$$ 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.