  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.
