8 problems

The patch-counting lower-bound conjecture. For every dimension $n\geq 1$ and Delone constants $(r,R)$, there is a positive constant $c=c(n,r,R)$ such that every aperiodic Delone se…

The linear repetitivity conjecture. Every aperiodic linearly repetitive Delone set is densely repetitive.

Let $X$ be a Delone set in $\mathbb{R}^d$ with parameters $(r,R)$, where $R$ is the radius of a largest empty ball, and let $\hat{\rho}_d=\hat{\rho}_d(r,R)$ be the smallest radius…

Let $X\subset\mathbb R^3$ be a Delone set with mutually equivalent $2R$-clusters. For each point $x\in X$, let $S_x(2R)$ denote its local group. Weak 5-gonal symmetry conjecture. T…

Let $X\subset\mathbb R^3$ be a Delone set with parameter $R$. For each point $x\in X$, let $S_x(2R)$ be its local group, and let $Y$ be the subset of points whose local groups cont…

Let $X\subset\mathbb R^3$ be a Delone set with parameter $R$. For each point $x\in X$, let $S_x(2R)$ be its local group, and let $K$ be the subset of points whose local groups cont…

Let $\varLambda\subset\mathbb R^3$ be a Delone set. A set of $\varLambda$-directions is in general position when any three directions from it span the whole $3$-space. Seven-direct…

Let $n\ge 3$, let $N=[n,2]$, and let $m_n$ denote the minimum number of pairwise nonparallel $\varLambda$-directions whose X-rays determine every convex subset of a strong $n$-cycl…