On $\mathbb F_2^\omega $-affine-exchangeable probability measures
Volume 279 / 2024
Abstract
In relation to a problem of Aldous in exchangeability theory, and in connection with arithmetic combinatorics and ergodic theory, we characterize strongly stationary systems in characteristic 2 (we call them affine-exchangeable measures). Such systems for $\mathbb Z$ actions were introduced by Furstenberg and Katznelson and characterized by Frantzikinakis. Let $\mathcal P(\mathrm B)$ denote the space of Borel probability measures on any standard Borel space B. Affine-exchangeable systems are probability measures on product spaces indexed by the vector space $\mathbb F_2^\omega $ (i.e. measures in $\mathcal P(\mathrm B^{\mathbb F_2^{\omega }})$) that are invariant under the coordinate permutations induced by all affine automorphisms of $\mathbb F_2^{\omega }$. We describe the extreme points of the space of such affine-exchangeable measures. We prove that there is a single structure underlying every such measure, namely, a random infinite-dimensional cube (sampled using Haar measure adapted to a specific filtration) on a countable power of the 2-adic integers. Indeed, every extreme affine-exchangeable measure is obtained from a $\mathcal P(\mathrm B)$-valued function on this group, by a vertex wise composition with this random cube. As a consequence we prove that the convex set of affine-exchangeable measures equipped with the vague topology is a Bauer simplex. We also obtain a correspondence between affine-exchangeability and limits of convergent sequences of (compact-metric-space valued) functions on vector spaces $\mathbb F_2^n$ as $n\to \infty $, thus establishing the above-mentioned group as a general limit domain valid for any such sequence. In particular, we obtain a complete limit theory for the densities of linear forms (of arbitrary complexity) in functions of finite abelian groups of exponent 2. Such arithmetic limit theories were only known for linear forms with restricted structure.
