Isoperimetric stability in lattices

Abstract

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