5 problems

Let $(X,T)$ be a topological dynamical system, and let $\mathcal{U}$ be a finite open cover of $X$. Define $p^*_{X,\mathcal{U}}(n)$ as the maximal, over all $n$-element subsets…

Let $\mathcal A$ be a finite alphabet and let $\eta\in\mathcal A^{\mathbb Z^2}$ be a configuration. For a non-empty set $\mathcal U\subset\mathbb Z^2$, the restriction…

Let $etafrak ainmathcal{A}^{mathbb{Z}^2}$ be a configuration with $mathcal{A}subsetmathbb{Z}$, and suppose … is a $\mathbb{Z}$-minimal periodic decomposition. Let…

Let $mathcal{S}subsetmathbb{Z}^2$ be a finite convex lattice set, meaning that its convex hull $operatorname{Conv}(mathcal{S})subsetmathbb{R}^2$ is closed and … For a configurati…

Let $c\in A^{\mathbb{Z}^d}$ be a configuration over a finite alphabet $A$. For a finite domain $D\subseteq\mathbb{Z}^d$, let $P_c(D)$ denote the number of distinct $D$-patterns in…