Strong marker sets and applications

We prove the existence of clopen marker sets with some strong regularity property. For each n ≥ 1 n\geq 1 and any integer d ≥ 1 d\geq 1 , we show that there are a positive integer D D and a clopen marker set M M in F ( 2 Z n ) F(2^{\mathbb {Z}^n}) such that for any distinct x , y ∈ M x,y\in M in the same orbit, ρ ( x , y ) ≥ d \rho (x,y)\geq d , where ρ ( x , y ) \rho (x,y) is the ℓ ∞ \ell ^\infty -norm of the unique g ∈ Z n g\in \mathbb {Z}^n such that g ⋅ x = y g\cdot x=y ; for any