The structure of totally disconnected Host--Kra--Ziegler factors, and the inverse theorem for the $U^k$ Gowers uniformity norms on finite abelian groups of bounded torsion

Let $\mathrm{\Gamma }$ be a countable abelian group, let $k\ge 1$, and let $\mathrm{X}=(X,\mathcal{X},\mu ,T)$ be an ergodic $\mathrm{\Gamma }$-system of order $k$ in the sense of Host–Kra–Ziegler. The $\mathrm{\Gamma }$-system $\mathrm{X}$ is said to be totally disconnected if all its structure groups are totally disconnected. We show that any totally disconnected $\mathrm{\Gamma }$-system of order $k$ is a generalized factor of a ${\mathbb{Z}}^{\omega }$-system with the structure of a Weyl system. As a consequence of this structure theorem, we show that totally disconnected $\mathrm{\Gamma }$-systems of order $k$ are represented by translations on double cosets of nilpotent Polish groups. By a correspondence principle of two of us, we can use this representation to establish a (weak) inverse theorem for the $U^k$ Gowers uniformity norms on finite abelian groups of bounded torsion.