Isoperimetric stability in lattices

We obtain isoperimetric stability theorems for general Cayley digraphs on ${\mathbb{Z}}^d$. For any fixed $B$ that generates ${\mathbb{Z}}^d$ over $\mathbb{Z}$, we characterise the approximate structure of large sets $A$ that are approximately isoperimetric in the Cayley digraph of $B$: we show that $A$ must be close to a set of the form $kZ\cap {\mathbb{Z}}^d$, where for the vertex boundary $Z$ is the conical hull of $B$, and for the edge boundary $Z$ is the zonotope generated by $B$.