${\mathbb{Z}}_k$-smooth approximation of mod $k$ integral cycles

We prove that every mod $k$ integral cycle $T$ of sufficiently high codimension in an oriented Riemannian manifold $\mathcal{M}$ can be approximated in flat norm by a cycle which is a ${\mathbb{Z}}_k$-smooth submanifold $\mathrm{\Sigma }$ of nearly the same area, up to a singular set of codimension 5. If in addition $T$ belongs to a mod $k$ homology class admitting an embedded ${\mathbb{Z}}_k$-smooth manifold representative, then $\mathrm{\Sigma }$ can be taken fully regular, up to the codimension one stratum of classical singularities. This extends the smooth approximation theory to cycles with coefficients in $\mathbb{Z}\mathrm{/}k\mathbb{Z}$ in stable regimes. Moreover, we provide examples showing the impossibility of locally approximating minimizing cycles by smooth or ${\mathbb{Z}}_k$-smooth submanifolds, even in homology classes with smooth or ${\mathbb{Z}}_k$-smooth representatives.