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 $n, d_0\geq 1$ 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 $\Delta$ such that (1) for any distinct $x,y\in M$ in the same orbit, $\rho(x,y)\geq d_0$; (2) for any $v\in S$ and any $x\in F(2^{\mathbb{Z}^n})$, there are non-negative integers $a,b\leq \Delta$ such that $av\cdot x\in M$ and $-bv\cdot x\in M$. Here $\rho$ denotes the Euclidean metric. The same result then holds for the standard supremum-norm metric $\rho_\infty$ (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.
From: Su Gao [view email]
[v1]
Sun, 14 Jun 2026 06:41:30 UTC (21 KB)
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