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

TL;DR

This paper proves that for certain sets in the plane, there exists a point in one set such that the set of distances from that point to another set has positive Lebesgue measure, advancing the distance set problem.

Contribution

It establishes the regular case of the distance set problem in the plane using multi-scale analysis and Mizohata-Takeuchi-type estimates.

Findings

Existence of a point y in F with positive measure distance set

Resolution of the regular case of the distance set problem in the plane

Development of multi-scale Mizohata-Takeuchi estimates with small power-loss

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.