Mean dimension and a sharp embedding theorem: extensions of aperiodic subshifts

We show that if $(X,T)$ is an extension of an aperiodic subshift (a subsystem of $({1,2,\mathrm{.}\mathrm{.}\mathrm{.},l}^{\mathbb{Z}},\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t})$ for some $l\in \mathbb{N}$) and has mean dimension $mdim(X,T)<\frac{D}{2}$ $(D\in \mathbb{N}$), then it embeds equivariantly in (([0,1]^{D})^{$\mathbb{Z}$},$\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t}$)$. The result is sharp. If $(X,T)$ is an extension of an aperiodic zero-dimensional system then it embeds equivariantly in $(([0,1{]}^{D+1}{)}^{\mathbb{Z}},\mathrm{s}\mathrm{h}\mathrm{i}\mathrm{f}\mathrm{t})$.