Given a positive integer N and real number α ∈ [0, 1], let m(α, N ) denote the minimum, over all sets A ⊆ Z N of size at least αN , of the normalized count of 3-term arithmetic progressions contained in A. A theorem of Croot states that m(α, N ) converges as N → ∞ through the primes, answering a question of Green. Using recent advances in higher-order Fourier analysis, we prove an extension of this theorem, showing that the result holds for k-term progressions for general k and further for all systems of integer linear forms of finite complexity. We also obtain a similar convergence result for the maximum densities of sets free of solutions to systems of linear equations. These results rely on a regularity method for functions on finite cyclic groups that we frame in terms of periodic nilsequences, using in particular some regularity results of Szegedy (relying on his joint work with Camarena) and the equidistribution results of Green and Tao.

The work toward these main theorems yields several other results of inherent interest, including the following: endowing every compact nilspace with a Borel probability measure that generalizes the Haar measure on compact abelian groups (Proposition 2.2.5); an automatic continuity result for Borel morphisms between compact nilspaces (Theorem 2.4.6); a rigidity result for morphisms into compact nilspaces of finite rank (Theorem 2.8.2). Several of the main tools used to obtain these results rely on the theory of continuous systems of measures, which provides a natural framework for measure theoretic aspects of compact nilspaces. This is detailed in Section 2.2.An alternative treatment of compact nilspaces is given by Gutman, Manners, and Varjú in the series of papers [20,21,22].Let us end this introduction by evoking one of the central motivations for the study of compact nilspaces. This motivation concerns the analysis of uniformity norms. These norms were introduced in arithmetic combinatorics by Gowers [15], and independently in ergodic theory (as the analogous uniformity seminorms) by Host and Kra [25]. The uniformity norms, or U d norms (one such norm for each integer d ≥ 2), are defined on the space of complex-valued functions on a finite (or more generally a compact) abelian group. These norms provide useful tools to control averages of bounded functions over certain linear configurations in abelian groups, configurations such as arithmetic progressions (such control can be obtained via the so-called generalized Von Neumann theorems); see [17] and [18]. A major topic concerning these norms is the analysis, for each d ≥ 2, of the harmonics that determine whether the U d norm of a function is small or large. For the U 2 norm these characteristic harmonics are simply the characters from Fourier analysis. For d > 2, this topic leads to the theory of higher order Fourier analysis. Central to this theory are the results known as inverse theorems for the U d norms, proved by Green, Tao and Ziegler [19], and independently by Szegedy [39]. Essentially, these theorems tell us that, for the U d norm of functions on Z/NZ, one can use as characteristic harmonics the functions known as (d − 1)-step nilsequences. In the approach to these theorems developed by Szegedy (consisting principally in [39] and his work with Camarena [9]), compact nilspaces play a fundamental role. Indeed, roughly speaking, in this approach a nilsequence on Z/NZ is obtained as a composition F •ϕ, where ϕ is a nilspace morphism from Z/NZ to a compact nilspace X, and F is some continuous function X → C. In this way, compact nilspaces play a role in higher order Fourier analysis that generalizes the role played by the circle group R/Z in the definition of characters in Fourier analysis. In these notes we shall concentrate on the theory of compact nilspaces in itself. For further introduction to higher order Fourier analysis and its applications, we refer the reader to the survey [16]. DISCRETE ANALYSIS, 2017:16, 57pp. DISCRETE ANALYSIS, 2017 Lemma 2.1...

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We study a class of measure-theoretic objects that we call cubic couplings, on which there is a common generalization of the Gowers norms and the Host-Kra seminorms. Our main result yields a complete structural description of cubic couplings, using nilspaces. We give three applications. Firstly, we describe the characteristic factors of Host-Kra type seminorms for measure-preserving actions of countable nilpotent groups.This yields an extension of the structure theorem of Host and Kra. Secondly, we characterize sequences of random variables with a property that we call cubic exchangeability.These are sequences indexed by the infinite discrete cube, such that for every integer k ≥ 0 the joint distribution's marginals on affine subcubes of dimension k are all equal.In particular, our result gives a description, in terms of compact nilspaces, of a related exchangeability property considered by Austin, inspired by a problem of Aldous. Finally, using nilspaces we obtain limit objects for sequences of functions on compact abelian groups (more generally on compact nilspaces) such that the densities of certain patterns in these functions converge. The paper thus proposes a measure-theoretic framework on which the area of higher-order Fourier analysis can be based, and which yields new applications of this area in a unified way in ergodic theory and arithmetic combinatorics.

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.