Strong marker sets for arbitrary generating sets of ${\mathbb{Z}}^n$

Gao and Wang proved a strong clopen marker theorem for finite generating sets of ${\mathbb{Z}}^n$ under the assumption that each generator has support of size either $1$ or $n$. We show that this support assumption can be removed. The proof is a short conjugacy argument: after a unimodular change of coordinates, any finite set of nonzero lattice vectors can be put in full-support position, allowing one to apply the theorem of Gao and Wang and conjugate the resulting marker set back.