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

Abstract

We classify measures on $\{0,1\}^{\mathbb{Z}^d}$, $d \geq 3$, 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.