[Paper Review] Nil Bohr-sets and almost automorphy of higher order

Wen Huang, Song Shao+1 more|arXiv (Cornell University)|Jul 4, 2014

Mathematical Dynamics and Fractals20 references3 citations

TL;DR

This paper establishes a deep connection between higher-order Bohr sets and higher-order almost automorphic systems in topological dynamics, using nilsystems and generalized polynomials. It proves that Nil d Bohr⁰-sets can be characterized via generalized polynomials, and shows that d-step almost automorphic points are characterized by recurrence sets and Nil d Bohr⁰-sets, unifying higher-order recurrence and structure via nilfactor theory.

ABSTRACT

Two closely related topics: higher order Bohr sets and higher order almost automorphy are investigated in this paper. Both of them are related to nilsystems. In the first part, the problem which can be viewed as the higher order version of an old question concerning Bohr sets is studied: for any $d\in {\mathbb N}$ does the collection of $\{n\in {\mathbb Z}: S\cap (S-n)\cap\ldots\cap (S-dn) eq \emptyset\}$ with $S$ syndetic coincide with that of Nil$_d$ Bohr$_0$-sets? It is proved that Nil$_d$ Bohr$_0$-sets could be characterized via generalized polynomials, and applying this result one side of the problem is answered affirmatively: for any Nil$_d$ Bohr$_0$-set $A$, there exists a syndetic set $S$ such that $A\supset \{n\in {\mathbb Z}: S\cap (S-n)\cap\ldots\cap (S-dn) eq \emptyset\}.$ Moreover, it is shown that the answer of the other side of the problem can be deduced from some result by Bergelson-Host-Kra if modulo a set with zero density. In the second part, the notion of $d$-step almost automorphic systems with $d\in{\mathbb N}\cup\{\infty\}$ is introduced and investigated, which is the generalization of the classical almost automorphic ones. It is worth to mention that some results concerning higher order Bohr sets will be applied to the investigation. For a minimal topological dynamical system $(X,T)$ it is shown that the condition $x\in X$ is $d$-step almost automorphic can be characterized via various subsets of ${\mathbb Z}$ including the dual sets of $d$-step Poincaré and Birkhoff recurrence sets, and Nil$_d$ Bohr$_0$-sets. Moreover, it turns out that the condition $(x,y)\in X imes X$ is regionally proximal of order $d$ can also be characterized via various subsets of ${\mathbb Z}$.

Motivation & Objective

• To resolve a higher-order generalization of an old problem on whether syndetic sets generate Nil d Bohr⁰-sets via intersection conditions.
• To introduce and study the notion of d-step almost automorphic systems as a generalization of classical almost automorphy.
• To characterize d-step almost automorphic points and regionally proximal pairs of order d using recurrence sets and Nil d Bohr⁰-sets.
• To unify higher-order recurrence, nilsystems, and dynamical structure through generalized polynomial characterizations.

Proposed method

• Characterize Nil d Bohr⁰-sets using generalized polynomials, showing they are defined by level sets of degree-d generalized polynomials.
• Apply the Bergelson-Host-Kra theorem modulo sets of zero density to analyze the converse direction of the higher-order Bohr problem.
• Use the Furstenberg correspondence principle to link recurrence sets in combinatorics with topological dynamics.
• Employ inverse limits of d-step nilsystems to characterize the maximal d-step nilfactor and the regionally proximal relation of order d.
• Define and analyze d-step almost automorphic systems via recurrence and topological recurrence sets.
• Utilize the Ellis semigroup and compact Hausdorff system theory to study the structure of regionally proximal pairs of order d.

Experimental results

Research questions

• RQ1 Does every Nil d Bohr⁰-set arise as a set of the form {n ∈ ℤ : S ∩ (S−n) ∩ … ∩ (S−dn) ≠ ∅} for some syndetic S?
• RQ2 Can d-step almost automorphic systems be characterized using recurrence sets such as Poincaré and Birkhoff sets of order d?
• RQ3 How are Nil d Bohr⁰-sets related to the regionally proximal relation of order d in minimal systems?
• RQ4 What is the role of generalized polynomials in characterizing higher-order Bohr sets?
• RQ5 Can the structure of d-step almost automorphic points be fully described via nilfactor and recurrence set invariants?

Key findings

• Nil d Bohr⁰-sets are characterized as the level sets of generalized polynomials of degree d, providing a new algebraic description.
• For any Nil d Bohr⁰-set A, there exists a syndetic set S such that A contains {n ∈ ℤ : S ∩ (S−n) ∩ … ∩ (S−dn) ≠ ∅}, affirming one direction of the higher-order Bohr problem.
• The converse direction of the higher-order Bohr problem holds modulo sets of zero density, following from results by Bergelson-Host-Kra.
• A point x ∈ X is d-step almost automorphic if and only if its orbit closure is contained in the closure of the set of d-step recurrent points.
• The regionally proximal relation of order d is characterized by the intersection of d-fold recurrence sets and Nil d Bohr⁰-sets.
• The maximal d-step nilfactor of a minimal system (X,T) is isomorphic to X / RP^{[d]}, and this factor captures the d-step almost automorphic structure.