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

2026/03/16 by Bochen Liu
Mathematics · #math.CA #math.AP #math.CO

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Abstract

Suppose E, F are Borel sets in the plane, dim_H E>1, dim_H E+dim_H F>2, and F has equal Hausdorff and packing dimension. We prove that there exists y∈ F such that the pinned distance set Δ_y(E):=\|x-y|:x∈ 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.

Citations

• A Counterexample to the Mizohata-Takeuchi Conjecture

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