15 problems

Period Conjecture. For each integer $j=1,\dots,d$, there is a positive constant $c_j(r,R)$ such that, if for some $t_0(X)$

The rank-of-periods conjecture. For each $j$ with $1\leq j\leq n$, there is a positive constant $c_j=c_j(n,r,R)$ such that, if

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 $D\subset\mathbb{R}^d$ be a Delone set. A Delone set is aperiodic if it has no nonzero translational period, linearly repetitive if every finite pattern occurring in it reappea…

Delone-set and overlapping-substitution conjecture. For an appropriate choice of $F$ and $A$, this set is a Delone set of $\Omega$ with inflation symmetry, and there is a correspon…

Let $S_{\mathbf{x}}(2R)$ be the group of isometries that stabilize both a point $\mathbf{x}$ of a $2R$-isometric Delone set in $\mathbb{R}^d$ and its $2R$-cluster. Let $h_d$ denote…

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 $X\subset\mathbb{R}^3$ be a Delone set with packing radius $r$, covering radius $R$, regularity radius $\widehat{R}$, and $N(2R)=1$. A local $5$-fold symmetry means that the…

Let $\Omega$ be the pattern space of a Delone set in $\mathbb{R}^d$, with $d\geq 1$, and let $\mathcal{H}$ denote the space of $L^2$-differential $1$-forms associated with the rele…

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…