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

Bochen Liu

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.

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

Abstract

Suppose are Borel sets in the plane, , , and has equal Hausdorff and packing dimension. We prove that there exists such that the pinned distance set 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.
Paper Structure (11 sections, 9 theorems, 127 equations)

This paper contains 11 sections, 9 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Falconer distance conjecture
  3. Mizohata-Takeuchi conjecture
  4. Preliminaries
  5. A multi-scale Good-Bad decomposition
  6. The single-scale Good-Bad decomposition of Guth-Iosevich-Ou-Wang
  7. A multi-scale Good-Bad decomposition
  8. Multi-scale Mizohata-Takeuchi-type estimates
  9. Proof of Theorem \ref{['thm-warmup']}
  10. $L^2$ estimates of the good part
  11. Proof of the main Theorem

Key Result

Theorem 1.1

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. Then there exists $y\in F$ such that the pinned distance set has positive Lebesgue measure. In particular, if $E\subset\mathbb{R}^2$ has equal Hausdorff and packing dimension $>1$, then $|\Delta_y(E)|>0$ for some $y\in E$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1: Lemma 3.5 in GIOW18
  • Proposition 3.2
  • Proposition 3.3
  • proof : Proof of Proposition \ref{['prop-good-bad']}
  • Proposition 4.1
  • Lemma 4.2
  • proof : Proof of Lemma \ref{['lem-rewrite-good-part']}
  • Lemma 4.3
  • ...and 3 more