Related papers: On exceptional sets of radial projections

We prove two conjectures in this paper. The first conjecture is by Lund, Pham and Thu: Given a Borel set $A\subset \mathbb{R}^n$ such that $\dim A\in (k,k+1]$ for some $k\in\{1,\dots,n-1\}$. For $0<s<k$, we have \[ \text{dim}(\{y\in…

We provide several new answers on the question: how do radial projections distort the dimension of planar sets? Let $X,Y \subset \mathbb{R}^{2}$ be non-empty Borel sets. If $X$ is not contained on any line, we prove that \[ \sup_{x \in X}…

We improve the best known lower bound for the dimension of radial projections of sets in the plane. We show that if $X,Y$ are Borel sets in $\R^2$, $X$ is not contained in any line and $\dim_H(X)>0$, then $$\sup\limits_{x\in X} \dim_H(\pi_x…

We generalize the recent results on radial projections by Orponen, Shmerkin, Wang using two different methods. In particular, we show that given $X,Y\subset \mathbb{R}^n$ Borel sets and $X\neq \emptyset$. If $\dim Y \in (k,k+1]$ for some…

This paper contains two results on the dimension and smoothness of radial projections of sets and measures in Euclidean spaces. To introduce the first one, assume that $E,K \subset \mathbb{R}^{2}$ are non-empty Borel sets with…

For $e \in S^{2}$, the unit sphere in $\mathbb{R}^3$, let $\pi_{e}$ be the orthogonal projection to $e^{\perp} \subset \mathbb{R}^{3}$, and let $W \subset \mathbb{R}^{3}$ be any $2$-plane, which is not a subspace. We prove that if $K…

A Borel set $B \subset \mathbb{R}^{n}$ is visible from $x \in \mathbb{R}^{n}$, if the radial projection of $B$ with base point $x$ has positive $\mathcal{H}^{n - 1}$ measure. I prove that if $\dim B > n - 1$, then $B$ is visible from every…

First, let $K \subset B(0,1) \subset \mathbb{R}^{2}$ be a set with $\mathcal{H}_{\infty}^{1}(K) \sim 1$, and write $\pi_{e}(K)$ for the orthogonal projection of $K$ into the line spanned by $e \in S^{1}$. For $1/2 \leq s < 1$, write $$E_{s}…

This paper is concerned with restricted families of projections in $\mathbb{R}^{3}$. Let $K \subset \mathbb{R}^{3}$ be a Borel set with Hausdorff dimension $\dim K = s > 1$. If $\mathcal{G}$ is a smooth and sufficiently well-curved…

Let $\gamma:[0,1]\rightarrow \mathbb{S}^{2}$ be a non-degenerate curve in $\mathbb{R}^3$, that is to say, $\det\big(\gamma(\theta),\gamma'(\theta),\gamma"(\theta)\big)\neq 0$. For each $\theta\in[0,1]$, let $V_\theta=\gamma(\theta)^\perp$…

We give upper bounds on the number of exceptional radial projections of arbitrary subsets of vector spaces over finite fields. Our bounds do not depend on the dimension of the ambient space. Let $\mathbb{F}_q^d$ be the $d$-dimensional…

We prove a version of the Erd\H{o}s--Beck Theorem from discrete geometry for fractal sets in all dimensions. More precisely, let $X\subset \mathbb{R}^n$ Borel and $k \in [0, n-1]$ be an integer. Let $\dim (X \setminus H) = \dim X$ for every…

Let $\gamma:[0,1]\rightarrow \mathbb{S}^{2}$ be a non-degenerate curve in $\mathbb{R}^3$, that is to say, $\det\big(\gamma(\theta),\gamma'(\theta),\gamma''(\theta)\big)\neq 0$. For each $\theta\in[0,1]$, let…

It is well known that if $A \subseteq \mathbb{R}^n$ is an analytic set of Hausdorff dimension $a$, then $\dim_H(\pi_VA)=\min\{a,k\}$ for a.e.\ $V\in G(n,k)$, where $G(n,k)$ denotes the set of all $k$-dimensional subspaces of $\mathbb{R}^n$…

For any $x\in\mathbb{R}^d$, $d\geq 2$, denote $\pi^x: \mathbb{R}^d\backslash\{x\}\rightarrow S^{d-1}$ as the radial projection $$\pi^x(y)=\frac{y-x}{|y-x|}. $$ Given a Borel set $E\subset{\Bbb R}^d$, $\dim_{\mathcal{H}} E\leq d-1$, in this…

Let $d \geq 2$ and $d - 1 < s < d$. Let $\mu$ be a compactly supported Radon measure in $\mathbb{R}^{d}$ with finite $s$-energy. I prove that the radial projections $\pi_{x\sharp}\mu$ of $\mu$ are absolutely continuous with respect to…

Let $0 \leq s \leq 1$, and let $\mathbb{P} := \{(t,t^{2}) \in \mathbb{R}^{2} : t \in [-1,1]\}$. If $K \subset \mathbb{P}$ is a closed set with $\dim_{\mathrm{H}} K = s$, it is not hard to see that $\dim_{\mathrm{H}} (K + K) \geq 2s$. The…

Let $0 \leq s \leq 1$. A set $K \subset \mathbb{R}^{2}$ is a Furstenberg $s$-set, if for every unit vector $e \in S^{1}$, some line $L_{e}$ parallel to $e$ satisfies $$\dim_{\mathrm{H}} [K \cap L_{e}] \geq s.$$ The Furstenberg set problem,…

Let $\{\pi_{e} \colon \mathbb{H} \to \mathbb{W}_{e} : e \in S^{1}\}$ be the family of vertical projections in the first Heisenberg group $\mathbb{H}$. We prove that if $K \subset \mathbb{H}$ is a Borel set with Hausdorff dimension…

We prove the following variant of Marstrand's theorem about projections of cartesian products of sets: Let $K_1,...,K_n$ Borel subsets of $\mathbb R^{m_1},... ,\mathbb R^{m_n}$ respectively, and $\pi:\mathbb R^{m_1}\times...\times\mathbb…