Regularity and inverse theorems for uniformity norms on compact abelian groups and nilmanifolds

Abstract We prove a general form of the regularity theorem for uniformity norms, and deduce an inverse these norms which holds class compact nilspaces including all abelian groups, also nilmanifolds; in particular we thus obtain first non-abelian versions such theorems. derive results from structure cubic couplings, thereby unifying with Host–Kra Ergodic Structure Theorem. A unification this kind had been propounded as conceptual prospect by Host Kra. Our work provides new on nilspaces. In particular, stability result nilspace morphisms. strengthen Gutman, Manners Varjú, proving that k -step finite rank is toral (in connected nilmanifold) if only its -dimensional cube set connected. morphism cyclic group prime order into finite-rank sufficiently balanced (i.e. equidistributed certain quantitative multidimensional sense), then toral. As application this, proof refinement Green–Tao–Ziegler theorem.