Last updated
2020-09-29T10:37:22.753+01:00
Abstract
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 $B$.
$\mathbb{Z}^d$. 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 $B$.
Symplectic ID
1123481
Download URL
http://arxiv.org/abs/2007.14457v2
Submitted to ORA
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Publication type
Journal Article