Nachi Avraham Re’em

How Periodic Can a Point Process Be in Groups Without Lattices? (Unusual room!)

Every probability preserving action of a locally compact group G on (X,m) admits a cross section Y (assuming countable stabilizers), which is a Borel set in X such that for every x ∈ X, the return times set
Y(x)={ g ∈ G : g.x ∈ Y }
is nonempty and locally finite. Hence, the random point set Y(x) forms a stationary point process on G. The system (G,X,Y,m) is "periodic" when Y(x) is a random coset of a lattice in G. The motivating question is: When G admits no lattice, how periodic can a system (G,X,Y,m) be?

To address this, I will first introduce a quantitative measure of periodicity using a higher-order version of Kac’s lemma. Then I will focus on a certain class of abelian groups, including p-adic groups and the finite adeles, to show that most periodic systems for such groups are precisely cut-and-project actions of the group coupled with ℝ. To illustrate this characterization concretely I will discuss the following application.

Theorem: Let P be a set of finite adeles such that P-P is uniformly discrete. Then d*(P−P)≥ 2 ⋅ d*(P) for the upper Banach density, with equality if and only if P-P contains, with full density, the difference set of a generalized Farey fractions set { q∈ℚ : q∈ [0,a] } - { q∈ℚ : q∈ [0,a] } embedded diagonally, or a dilation of such a set by a finite adelic integer.

Based on joint works with Michael Björklund and Rickard Cullman.

Note unusual room!