Finite-to-one equivariant maps and mean dimension

We show that a minimal dynamical system $(X,\mathbb{Z})$ on a compact metric $X$ with mdim$X=d$ admits for every natural $k>d$ an equivariant map to the shift $([0,1{]}^k{)}^{\mathbb{Z}}$ such that each fiber of this map contains at most $[k\mathrm{/}(k-d)]k\mathrm{/}(k-d)$ points.