Let $s \in [0,1]$. We show that a Borel set $N \subset \mathbb{R}^{2}$ whose every point is linearly accessible by an $s$-dimensional family of lines has Hausdorff dimension at most $2 - s$.
In this work we obtain a geometric characterization of the measures $\mu$ in $\mathbb{R}^{n+1}$ with polynomial upper growth of degree $n$ such that the $n$-dimensional Riesz transform $\mathcal{R}\mu (x) = \int \frac{x-y}{|x-y|^{n+1}}d\mu(y)$ …
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 a planar set E is …