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Joachim Cuntz, David Masser, Tobias Colding, Daniel Huybrechts, Jun-Muk Hwang, Olivier Schiffmann, Jakob Stix, Gábor Székelyhidi, Franz Stückle, Druck Und Verlag

Abstract

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

Locations

• Journal für die reine und angewandte Mathematik (Crelles Journal) - View - PDF