Strong marker sets for arbitrary generating sets

Abstract

We prove the existence of clopen strong marker sets in $F(2^{{\mathbb{Z}}^n})$ for arbitrary finite generating sets. Specifically, for any positive integers $n,d_0\ge 1$ and any finite generating set $S\subseteq {\mathbb{Z}}^n$, we construct a clopen set $M\subseteq F(2^{{\mathbb{Z}}^n})$ and a positive integer $\hat{I}”$ such that (1) for any distinct $x,y\in M$ in the same orbit, $\ddot{I}(x,y)\ge d_0$; (2) for any $v\in S$ and any $x\in F(2^{{\mathbb{Z}}^n})$, there are non-negative integers $a,b\le \hat{I}”$ such that $av\cdot x\in M$ and $-bv\cdot x\in M$. Here $\ddot{I}$ denotes the Euclidean metric. The same result then holds for the standard supremum-norm metric $\ddot{I}{}_{\infty }$ (with an adjusted constant), by the equivalence of norms on ${\mathbb{Z}}^n$. The proof introduces polyhedral packages in ${\mathbb{R}}^n$ as a generalization of the rectangular packages used in earlier work, enabling the construction to handle generating vectors with arbitrary coordinate patterns. As an application, we obtain a continuous proper edge $(2|S|+1)$-coloring of the Schreier graph on $F(2^{{\mathbb{Z}}^n})$ with generating set $S$, recovering a result of Gao--Wang--Wang.