8 problems

Let $d\geq 1$ and define the cone … For a set $A\subseteq\mathbb{R}^{d+1}$, its Fourier dimension is … where $\mathcal{P}(A)$ is the set of Borel probability measures on…

Let $\mathcal{C}$ be a compact family of closed cones. Suppose that at each point of a set $S$, each blow-up of $S$ is contained in some element of $\mathcal{C}$. Let $d$ be the la…

Let $C$ be a closed cone in $\mathbb{R}^n$. Its building dimension is … where $\operatorname{diff}(S):=\{x-y:x,y\in S\}$. White's building-dimension conjecture. The building dimens…

Optimal-band bound conjecture. The optimal upper bound for the number of bands in $(X,K)$ is

Let $\mathcal{C}$ denote the cone. Let $S$ be either the group $\mathcal{C}_2$ representing a two-fold rotation about an axis perpendicular to the $z$-axis, or the group…

Let $K$ be a cone in $\boldsymbol{Z}^d$, and consider a typical random walk on $\boldsymbol{Z}^d$ with nonzero drift, killed when it leaves $K$. A quasistationary distribution (QSD…

Let $K$ be a cone, let $S(n)$ be a random walk with increments $X$, and let … be its exit time from $K$ when started at $x\in K$. Suppose that $X=R\xi$, where $R$ is a non-negative…

Let $C_\alpha S^{2p-1}_{l\sin\alpha}$ be the cone of angle $\alpha$ and length $l>0$ over the odd-dimensional sphere $S^{2p-1}$, equipped with the standard metric induced by the im…