Examples of $d-$sets with irregular projection of Hausdorff measures

TL;DR

This paper constructs specific sets in Euclidean space with finite Hausdorff measure whose projections have Radon-Nikodym derivatives that are not in any L^p space for p>1, highlighting irregular projection properties.

Contribution

It provides explicit examples of sets with finite Hausdorff measure exhibiting irregular projection behavior in terms of Radon-Nikodym derivatives.

Findings

Proves existence of sets with positive finite Hausdorff measure and irregular projections.

Shows Radon-Nikodym derivatives are not in L^p for any p>1.

Highlights complexity of measure projections in geometric measure theory.

Abstract

Given positive integers $\ell<n$ and a real $d\in (\ell,n)$, we construct sets $K\subset \mathbb R^n$ with positive and finite Hausdorff $d-$measure such that the Radon-Nikodym derivative associated to all projections on $\ell-$dimensional planes is not an $L^p$ function, for all $p>1$.