Nil Bohr$_0$-sets, Poincar\'e recurrence and generalized polynomials

TL;DR

This paper characterizes Nil$_d$ Bohr$_0$-sets using generalized polynomials and explores their relationship with syndetic sets and higher order recurrence, linking combinatorial and dynamical properties.

Contribution

It proves Nil$_d$ Bohr$_0$-sets can be characterized via generalized polynomials and establishes their connection with syndetic sets and higher order recurrence.

Findings

Nil$_d$ Bohr$_0$-sets characterized by generalized polynomials

Existence of syndetic sets approximating Nil$_d$ Bohr$_0$-sets

Coincidence of collections in dynamical characterization of higher order almost automorphic points

Abstract

The problem which can be viewed as the higher order version of an old question concerning Bohr sets is investigated: for any $d\in \N$ does the collection of $\{n\in \Z: S\cap (S-n)\cap...\cap (S-dn)\neq \emptyset\}$ with $S$ syndetic coincide with that of Nil$_d$ Bohr$_0$-sets? In this paper it is proved that Nil$_d$ Bohr$_0$-sets could be characterized via generalized polynomials, and applying this result one side of the problem could be answered affirmatively: for any Nil$_d$ Bohr$_0$-set $A$, there exists a syndetic set $S$ such that $A\supset \{n\in \Z: S\cap (S-n)\cap...\cap (S-dn)\neq \emptyset\}.$ Note that other side of the problem can be deduced from some result by Bergelson-Host-Kra if modulo a set with zero density. As applications it is shown that the two collections coincide dynamically, i.e. both of them can be used to characterize higher order almost automorphic points.