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

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 $p>1$.

3 pages