Nil Bohr0-sets and polynomial recurrence

In Huang et al.  [17] it was proved that for any Nild  Bohr0-set AA, there are a minimal system (X,T)(X,T) and a non-empty open subset UU of XX with A⊃{n∈Z:U∩T−nU∩⋯∩T−dnU≠0̸}A⊃{n∈Z:U∩T−nU∩⋯∩T−dnU≠0̸}, and for any minimal system (X,T)(X,T) and any open non-empty U⊂XU⊂X, the set {n∈Z:U∩T−nU∩⋯∩T−dnU≠0̸}{n∈Z:U∩T−nU∩⋯∩T−dnU≠0̸} is an almost Nild Bohr0-set. The polynomial form of this problem is considered in this paper. It is shown that the latter is still true in the polynomial case, while the former is not in general. We also consider the special case when the system is a nilsystem.