8 research outputs found
Regularity and inverse theorems for uniformity norms on compact abelian groups and nilmanifolds
We prove a general form of the regularity theorem for uniformity norms, and
deduce a generalization of the Green-Tao-Ziegler inverse theorem, extending it
to a class of compact nilspaces including all compact abelian groups and
nilmanifolds. We derive these results from a structure theorem for cubic
couplings, thereby unifying these results with the ergodic structure theorem of
Host and Kra. The proofs also involve new results on nilspaces. In particular,
we obtain a new stability result for nilspace morphisms. We also strengthen a
result of Gutman, Manners and Varju, 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 (a quantitative form of multidimensional
equidistribution), then the nilspace is toral.Comment: 35 page