${\mathrm{ASL}}_n(\mathbb{Z})$ invariant random subsets of ${\mathbb{Z}}^n$

Abstract

We classify measures on $\{0,1{\}}^{{\mathbb{Z}}^d}$, $d\ge 3$, the space of subsets of ${\mathbb{Z}}^d$, which are invariant under all affine special linear transformations. In other words, we classify simple point processes on ${\mathbb{Z}}^d$ whose law is invariant under affine special linear transformations. We show that every such process is built from a random equivariant polynomial together with independent random sampling, a higher-order generalisation of the cut-and-project method: a random polynomial map is drawn from a distribution invariant under a natural action of ${\mathrm{SL}}_d(\mathbb{Z})$, each site is then retained independently with a probability determined by a measurable function of the polynomial's value, and the classical cut-and-project construction is recovered in the degree-one case. As a corollary, when the underlying ${\mathbb{Z}}^d$-action is weakly mixing the measure must be a convex combination of Bernoulli shifts, in the spirit of de Finetti's theorem on exchangeable processes. Our theorem also makes precise how the Howe--Moore theorem fails for the pair $({\mathrm{ASL}}_d(\mathbb{Z}),{\mathrm{SL}}_d(\mathbb{Z}))$. Motivated by this classification, we formulate a conjecture for ${\mathrm{ASL}}_d(\mathbb{R})$-invariant point processes on ${\mathbb{R}}^d$, predicting that any such set decomposes into a Poisson part and a quasicrystal part. The proofs rely on the interaction between the Host--Kra theory of characteristic factors, Zimmer's theory of dynamical cocycles of simple Lie groups, and the dynamics of ${\mathrm{SL}}_d(\mathbb{Z})$-actions on homogeneous spaces.

44 pages; comments welcome!