A Quantification of a Besicovitch Nonlinear Projection Theorem via Multiscale Analysis

TL;DR

This paper extends the Besicovitch projection theorem to nonlinear curve projections, providing a quantitative analysis using multiscale techniques to show that nearly purely unrectifiable sets have very small Favard curve length.

Contribution

It introduces a quantitative version of the Besicovitch nonlinear projection theorem using multiscale analysis, expanding the theorem's applicability to nonlinear projections.

Findings

Favard curve length is very small for nearly purely unrectifiable sets

Multiscale analysis effectively quantifies nonlinear projection properties

Extension of the classical theorem to nonlinear curve projections

Abstract

The Besicovitch projection theorem states that if a subset $E$ of the plane has finite length in the sense of Hausdorff measure and is purely unrectifiable (so its intersection with any Lipschitz graph has zero length), then almost every orthogonal projection of $E$ to a line will have zero measure. In other words, the Favard length of a purely unrectifiable $1$-set vanishes. In this article, we show that when linear projections are replaced by certain nonlinear projections called curve projections, this result remains true. In fact, we go further and use multiscale analysis to prove a quantitative version of this Besicovitch nonlinear projection theorem. Roughly speaking, we show that if a subset of the plane has finite length in the sense of Hausdorff and is nearly purely unrectifiable, then its Favard curve length is very small. Our techniques build on those of Tao, who in [Tao09] proves a quantification of the original Besicovitch projection theorem.