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Structure of sets with nearly maximal Favard length

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

How much can heavy lines cover?

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 …

Visible parts and slices of Ahlfors regular sets

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 …

An $\alpha$-number characterization of $L^{p}$ spaces on uniformly rectifiable sets

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

Necessary condition for the $L^2$ boundedness of the Riesz transform on Heisenberg groups

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 …

Integrability of orthogonal projections, and applications to Furstenberg sets

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

Cones, rectifiability, and singular integral operators

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 …

Two examples related to conical energies

In a recent article we introduced and studied conical energies. We used them to prove three results: a characterization of rectifiable measures, a characterization of sets with big pieces of Lipschitz graphs, and a sufficient condition for …

Sufficient condition for rectifiability involving Wasserstein distance $W_2$

A Radon measure $\mu$ is $n$-rectifiable if it is absolutely continuous with respect to $\mathcal{H}^n$ and $\mu$-almost all of supp$\mu$ can be covered by Lipschitz images of $\mathbb{R}^n$. In this paper we give two sufficient conditions for …

Necessary condition for rectifiability involving Wasserstein distance $W_2$

A Radon measure $\mu$ is $n$-rectifiable if it is absolutely continuous with respect to $n$-dimensional Hausdorff measure and $\mu$-almost all of supp$\mu$ can be covered by Lipschitz images of $\mathbb{R}^n$⁠. In this paper, we give a necessary …