Host-Kra factors for $\bigoplus_{p\in P}\mathbb{Z}/p\mathbb{Z}$ actions and finite dimensional nilpotent systems

The inverse theorem for the $U^3$ Gowers uniformity norm on arbitrary finite abelian groups: Fourier-analytic and ergodic approaches

“…Note in the above theorem that H is merely locally compact rather than a Lie group. This is necessary in order for the theorem to hold; see the example presented after [29, Conjecture 2.14] (in the discussion of [29,Theorem 4.3]). In particular, we cannot necessarily take H/Γ to be a nilmanifold, although we shall later see that it is still an inverse limit of nilmanifolds (in the category of compact nilspaces, not the category of Z ω -systems).…”

The structure of arbitrary Conze-Lesigne systems

“…The implication of (ii) from (i) (in both high and low characteristic) is [4,Lemma A.35 p∈P Z/pZ for some countable multiset P of primes, then X is the inverse limit of translational systems G n /Λ n , where each G n is nilpotent of class at most two. (ii) [35,Theorem 2.3]…”

“…(iii) [35,Theorem 2.10] If Γ = p∈P Z/pZ for some countable multiset P of primes, then there exists a natural number m = m(k) depending only on k, and an m-extension 7 Y of X, which is an ergodic separable Γ ′ -system for some countable abelian group Γ ′ which is the inverse limit of translational Γ ′ -systems G n /Λ n , where each G n is a finite dimensional 8 locally compact group of nilpotency class at most k, and Λ n is totally disconnected.…”

“…Main result. We now come to the main new result of this paper, which is to establish a complete description of Conze-Lesigne factors for arbitrary countable abelian groups Γ: See [35,Definition 2.6] for the definition of finite dimensionality for locally compact groups; we will not need this notion in the rest of this paper.…”

“…In view of Theorem 1.4, one could ask whether one could strengthen Theorem 1.7 further by requiring in (ii) that the G n are nilpotent Lie groups, and Λ n lattices, so that X would be the inverse limit of nilsystems. Unfortunately when Γ is not finitely generated, there are counterexamples that show that this stronger version of Theorem 1.7 fails; see the example presented after [35,Conjecture 2.14] (in the discussion of [35,Theorem 4.3]). In Theorem 1.7 it is not required that the system X be separable, but it turns out it is quite easy to reduce to this case, and indeed this will be one of the first steps in the proof.…”