Favard length and quantitative rectifiability

TL;DR

This paper establishes a quantitative link between Favard length and rectifiability for Ahlfors 1-regular sets, showing large Favard length implies the presence of Lipschitz graph pieces, advancing understanding of projection theorems and related conjectures.

Contribution

It provides a quantitative version of the Besicovitch projection theorem and addresses open questions in geometric measure theory.

Findings

Large Favard length implies the set contains a big Lipschitz graph piece.

Answers to questions of David, Semmes, Peres, and Solomyak.

Progress on Vitushkin's conjecture.

Abstract

The Favard length of a Borel set $E\subset\mathbb{R}^2$ is the average length of its orthogonal projections. We prove that if $E$ is Ahlfors 1-regular and it has large Favard length, then it contains a big piece of a Lipschitz graph. This gives a quantitative version of the Besicovitch projection theorem. As a corollary, we answer questions of David and Semmes and of Peres and Solomyak. We also make progress on Vitushkin's conjecture.