Quantitative Besicovitch projection theorem for irregular sets of directions

TL;DR

This paper establishes a quantitative version of the Besicovitch projection theorem for AD-regular sets, showing that large measure sets of directions with bounded projections imply the set is mostly contained in a Lipschitz graph, even when the directions are merely measurable.

Contribution

It extends previous results by allowing the set of good directions to be merely measurable and large in measure, not necessarily an arc, providing a broader applicability.

Findings

A large measure set of directions with bounded projections implies the set is mostly Lipschitz graph.

The result generalizes classical theorems by relaxing the regularity condition on the set of directions.

Provides new insights into the structure of AD-regular sets with respect to their projections.

Abstract

The classical Besicovitch projection theorem states that if a planar set $E$ with finite length is purely unrectifiable, then almost all orthogonal projections of $E$ have zero length. We prove a quantitative version of this result: if $E\subset\mathbb{R}^2$ is AD-regular and there exists a set of direction $G\subset \mathbb{S}^1$ with $\mathcal{H}^1(G)\gtrsim 1$ such that for every $\theta\in G$ we have $\|\pi_\theta\mathcal{H}^1|_E\|_{L^{\infty}}\lesssim 1$, then a big piece of $E$ can be covered by a Lipschitz graph $\Gamma$ with $\mathrm{Lip}(\Gamma)\lesssim 1$. The main novelty of our result is that the set of good directions $G$ is assumed to be merely measurable and large in measure, while previous results of this kind required $G$ to be an arc. As a corollary, we obtain a result on AD-regular sets which avoid a large set of directions, in the sense that the set of directions they span has a large complement. It generalizes the following easy observation: a set $E$ is contained in some Lipschitz graph if and only if the complement of the set of directions spanned by $E$ contains an arc.