Notes on compact nilspaces

Abstract

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...

Directional dynamical cubes for minimal -systems

“…From (6) we deduce that (v k+1 ε : ε ∈ {0, 1} d , ε k+1 = 1) = (u k ε : ε ∈ {0, 1} d , ε k+1 = 0) . By Lemma 12, we can glue the (k + 1)-th lower face of v k+1 with the (k + 1)-th upper face of u k and obtain the point u k+1 that belongs to Q T1,...,T d (X).…”

“…Note that if ε k+1 = 1 (or equivalently k + 1 ∈ ε) then, (6) v k+1 ε = x ε∪{k+2,...,d} = x (ε\{k+1})∪{k+1,k+2,...,d} = u k ε\{k+1} , where in the last equality we used (5) with ε \ {k + 1}, which is different from [d] \ {1} and [d].…”

The structure theory of nilspaces I

“…These notions of "cocycles" and "cohomology" are useful tools, and we will allude to them again below. For the relevant formal definitions of cocycles and coboundaries, see [GMV16a, Definition 4.8] and the subsequent discussion; for a more in-depth discussion of cohomology more generally, and its relation to extensions of cubespaces, see [ACS12, Section 2.10] (or [Can17b,Can17a]).…”

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“…Unfortunately, it has proved to be extremely hard to check the correctness of the arguments in the three papers, which, if all the details can be completed and checked, would give a different and in some ways more natural proof of the inverse theorem. At the time of writing, various people are working to produce clearer and more complete versions of the argument [9,10,[34][35][36]: it seems likely that Szegedy's ideas are fundamentally correct and that this is indeed an interesting alternative approach.…”