Strong marker sets and applications

We prove the existence of clopen marker sets with some strong regularity property. For each $n\geq 1$ and any integer $d\geq 1$, we show that there are a positive integer $D$ and a clopen marker set $M$ in $F(2^{\mathbb{Z}^n})$ such that (1) for any distinct $x,y\in M$ in the same orbit, $\rho(x,y)\geq d$; (2) for any $1\leq i\leq n$ and any $x\in F(2^{\mathbb{Z}^n})$, there are non-negative integers $a, b\leq D$ such that $a\cdot x\in M$ and $-b\cdot x\in M$. As an application, we obtain a clopen tree section for $F(2^{\mathbb{Z}^n})$. Based on the strong marker sets, we get a quick proof that there exist clopen continuous edge $(2n+1)$-colorings of $F(2^{\mathbb{Z}^n})$. We also consider a similar strong markers theorem for more general generating sets. In dimension 2, this gives another proof of the fact that for any generating set $S\subseteq \mathbb{Z}^2$, there is a continuous proper edge $(2|S|+1)$-coloring of the Schreier graph of $F(2^{\mathbb{Z}^n})$ with generating set $S$.