Cube structures of the universal minimal system, nilsystems and applications

We propose and develop an approach to studying nilsystems and their proximal extensions using cube structures associated with the universal minimal system. We provide alternative proofs for results regarding saturation properties of factor maps to maximal nilfactors in cubes, as well as new results and applications of independent interest to the structural theory of topological systems. In particular, we give a new proof that RP[d] is an equivalence relation, building upon the distal case, by establishing a description of this relation in algebraic terms. This is new even for d=1.