12 problems

Let $X\subset\mathbb{R}^d$ be a Borel set, with $\operatorname{dim_H}(X)\leq d-1$, that is not contained in a hyperplane. For $y\in X$, let … be the radial projection of $X$ from…

Let $Y\subset\mathbb{R}^n$ be Borel, and for $x\in\mathbb{R}^n$ let $\pi_x(Y)$ denote the radial projection of $Y\setminus\{x\}$ from $x$ onto the unit sphere. Liu's conjecture. ……

Let $E\subset \mathbb{R}^d$ be a Borel set, let $\dim_H(E)>d-1$, and let $s\leq d-1$ be a real number. Radial projection exceptional-set conjecture. … This conjecture strengthens t…

Let $A\subset\mathbb{R}^2$ be Borel. Let $\pi_0(A\setminus\{0\})$ be the radial projection from the origin, and let $E_+(A)$ denote the corresponding positive-dimensional exception…

Orponen–Shmerkin–Wang's conjecture. The hypothesis $\dim Y>k-1/k-\eta$ in the radial projection theorem should be replaceable by the weaker hypothesis

Sharp lower-bound conjecture. If $\dim_{\mathrm{H}}Y>k-1$, then

Let $p$ be prime. Let $E\subset\mathbb{F}_p^2$ satisfy $|E|\ll p$ and suppose that no line contains more than $c|E|$ points of $E$, for the constant $c$ from the conjecture. Finite…

Finite-field radial projection exceptional-set conjecture.

Let $E\subset \mathbb{R}^d$ be a Borel set with $\dim_H(E)\in(k-1,k]$ for some $k\in\{0,1,\ldots,d-1\}$. Liu–Orponen's radial projection conjecture. … This strengthens the exceptio…

Let $E$ and $F$ be Borel sets in the plane with $\dim_{\mathcal{H}}E>0$ and $\dim_{\mathcal{H}}F>0$, and suppose that $F$ is not contained in a line. For $y$ in the plane, let…

Radial projection exceptional-set conjecture. One should have

Let $K\subset\mathbb{R}^{2}$ be a Borel set, not contained on a line, and define the set of unit vectors spanned by $K$ by … Here $\dim_{\mathrm{H}}$ denotes Hausdorff dimension. D…