Towers for commuting endomorphisms, and combinatorial applications

We give an elementary proof of a generalization of Rokhlin's lemma for commuting non-invertible measure-preserving transformations, and we present several combinatorial applications.Comment: 13 pages. Referee's comments incorporated. To appear in Annales de l'Institut Fourie

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

On linear configurations in subsets of compact abelian groups, and invariant measurable hypergraphs

We prove an arithmetic removal result for all compact abelian groups, generalizing a finitary removal result of Kr\'al', Serra and the third author. To this end, we consider infinite measurable hypergraphs that are invariant under certain group actions, and for these hypergraphs we prove a symmetry-preserving removal lemma, which extends a finitary result of the same name by the second author. We deduce our arithmetic removal result by applying this lemma to a specific type of invariant measurable hypergraph. As a direct application, we obtain the following generalization of Szemer\'edi's theorem: for any compact abelian group $G$, any measurable set $A\subset G$ with Haar probability $\mu(A)\geq\alpha>0$ satisfies $$\int_G\int_G\; 1_A\big(x\big)\; 1_A\big(x+r\big) \cdots 1_A\big(x+(k-1)r\big) \; d\mu(x) d\mu(r) \geq c,$$ where the constant $c=c(\alpha,k)>0$ is valid uniformly for all $G$. This result is shown to hold more generally for any translation-invariant system of $r$ linear equations given by an integer matrix with coprime $r\times r$ minors.Comment: 36 pages. Minor changes. To appear in Annals of Combinatoric