Nil Bohr$_0$-sets, Poincaré recurrence and generalized polynomials

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.