A refined structure theorem for polynomial return-time sets in minimal systems

We prove a refinement of a recent structural result in topological dynamics due to Glasscock, Koutsogiannis, Le, Moreira, Richter, and Robertson, who showed that in a minimal system, polynomial return-times differ by non-piecewise syndetic sets from those in its maximal infinite-step pronilfactor. In particular, we prove that the infinite-step pronilfactor can be replaced by the maximal $k$-step pronilfactor of the system, where $k$ depends only on the given polynomials, with $k-1$ equal to the Host-Kra complexity of the polynomials.