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The problem which can be viewed as the higher order version of an oldquestion concerning Bohr sets is investigated: for any $d\in \N$ does thecollection 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 characterizedvia generalized polynomials, and applying this result one side of the problemcould be answered affirmatively: for any Nil$_d$ Bohr$_0$-set $A$, there existsa 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 deducedfrom some result by Bergelson-Host-Kra if modulo a set with zero density. Asapplications it is shown that the two collections coincide dynamically, i.e.both of them can be used to characterize higher order almost automorphicpoints.

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