4 problems

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…

Let $\Omega\subset\mathbb R^d$ be a set of finite, positive measure. A translational tiling of $\mathbb R^d$ is a collection of translates of $\Omega$ covering almost every point e…

A tiling of the plane is a covering of $\mathbb{R}^2$ by closed subsets, called tiles, whose distinct interiors are disjoint. A finite set of prototiles is aperiodic if it can form…