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Our main result marks progress on an old conjecture of Vitushkin. We show that a compact set in the plane with plenty of big projections (PBP) has positive analytic capacity, along with a quantitative lower bound. A higher dimensional counterpart is …

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 …

We show that for any compact set $E\subset {\mathbb{R}}^d$ the visible part of $E$ has Hausdorff dimension at most $d-1/6$ for almost every direction. This improves recent estimates of Orponen and Matheus. If $E$ is $s$-Ahlfors regular, where $sd-1$, we …

Let $E\subset B(1)\subset {\mathbb{R}}^2$ be an ${\mathcal{H}}^1$ measurable set with ${\mathcal{H}}^1(E)0$.

One formulation of Marstrand's slicing theorem is the following. Assume that $t\in (1,2]$, and $B\subset {\mathbb{R}}^2$ is a Borel set with ${\mathcal{H}}^t(B)t-1$? A positive answer for $t$-regular sets $B\subset {\mathbb{R}}^2$ was …

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 $\mathcal{R}\mu (x)=\int \frac{x-y}{|x-y{|}^{n+1}}d\mu (y)$ …

We give a characterization of $L^p(\sigma )$ for uniformly rectifiable measures $\sigma$ using Tolsa's $\alpha$-numbers, by showing, for $1

Let $\mu$ be a Radon measure on the $n$-th Heisenberg group ${\mathbb{H}}^n$. In this note we prove that if the $(2n+1)$-dimensional (Heisenberg) Riesz transform on ${\mathbb{H}}^n$ is $L^2(\mu )$-bounded, and if $\mu (F)=0$ for all Borel sets with …

Let $\mathcal{G}(d,n)$ be the Grassmannian manifold of $n$-dimensional subspaces of ${\mathbb{R}}^d$, and let ${\pi }_V:{\mathbb{R}}^d\to V$ be the orthogonal projection. We prove that if $\mu$ is a compactly supported Radon measure on …

Let $\mu$ be a Radon measure on ${\mathbb{R}}^d$. We define and study conical energies $\mathcal{E}(x,V,\alpha )$, which quantify the portion of $\mu$ lying in the cone with vertex $x\in {\mathbb{R}}^d$, direction $V\in G(d,d-n)$, and aperture $\alpha\in …