A quantitative version of the Besicovitch projection theorem via multiscale analysis

TL;DR

This paper develops a multiscale analysis approach to quantitatively refine the Besicovitch projection theorem, providing explicit bounds and characterizations of unrectifiable sets and their projections in the plane.

Contribution

It introduces a quantitative framework for the Besicovitch projection theorem using multiscale analysis, including explicit bounds on projections of Cantor sets.

Findings

Almost every projection of a purely unrectifiable set has measure zero.

Any planar set with two measure-zero projections is purely unrectifiable.

Provided an explicit upper bound on the average projection of a product Cantor set.

Abstract

By using a multiscale analysis, we establish quantitative versions of the Besicovitch projection theorem (almost every projection of a purely unrectifiable set in the plane of finite length has measure zero) and a standard companion result, namely that any planar set with at least two projections of measure zero is purely unrectifiable. We illustrate these results by providing an explicit (but weak) upper bound on the average projection of the $n^{th}$ generation of a product Cantor set.