The structure of arbitrary Conze-Lesigne systems (Q6547225)

\textit{H. Furstenberg}'s ergodic approach to Szemerédi's theorem [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)], which used in part what is now called the Furstenberg-Zimmer structure theorem for measure-preserving systems, led to extensive work on convergence of non-conventional ergodic averages. In particular \textit{B. Host} and \textit{B. Kra} [Ann. Math. (2) 161, No. 1, 397--488 (2005; Zbl 1077.37002)] and \textit{T. Ziegler} [J. Am. Math. Soc. 20, No. 1, 53--97 (2007; Zbl 1198.37014)] showed mean convergence of a large class of non-conventional ergodic averages via a refinement of the Furstenberg-Zimmer structure theory identifying a finer hierarchy of characteristic factors of various orders, with the order 2 factor being the Conze-Lesigne algebra used in an earlier special case, see [\textit{J.-P. Conze} and \textit{E. Lesigne}, Publ. Inst. Rech. Math. Rennes 1987, No. 1, 1--31 (1987; Zbl 0654.28012)]. Part of the significance of this work is the identification of these factors as inverse limits of translations on nilmanifolds. From a different point of view \textit{W. T. Gowers} [Geom. Funct. Anal. 8, No. 3, 529--551 (1998; Zbl 0907.11005); Geom. Funct. Anal. 11, No. 3, 465--588 (2001); Erratum 11, No. 4, 869 (2001; Zbl 1028.11005)] found a Fourier analytic approach to Szemerédi's theorem, introducing the field of higher-order Fourier analysis and placing `inverse theorems' (relating certain norms introduced by Gowers to correlation with functions derived from nilmanifolds) at the centre of developments. \textit{B. Green} et al. [Ann. Math. (2) 176, No. 2, 1231--1372 (2012; Zbl 1282.11007)] made significant progress on inverse theorems for cyclic groups, which (for example) was applied to find the correct asymptotics for primes in arithmetic progression by \textit{B. Green} and \textit{T. Tao} [Ann. Math. (2) 171, No. 3, 1753--1850 (2010; Zbl 1242.11071)], but the full picture is not yet complete. Relating the Host-Kra-Ziegler structure theory to the Green-Tao-Ziegler inverse theorems remains an appealing vision. Indeed, {B. Host} and {B. Kra} [loc. cit.] characterize the order \(k\) factors using Host-Kra-Gowers semi-norms of order \(k+1\), giving an infinitary counterpart of the Gowers uniformity norms. \N\NHere and in a companion paper by the first and third authors [Discrete Anal. 48, Paper No. 11 (2023)] we see contributions to this circle of ideas, and in particular further insights into the analogy. Between the two papers a Host-Kra structure theorem for arbitrary abelian systems of order 2 is found, and used to give a qualitative proof of the inverse theorem for Gowers uniformity norms of order \(3\) for arbitrary finite abelian groups via a correspondence principle.