On the maximal spectral type of nilsystems (Q6621359)

Generalizing the work of \textit{B. Host} et al. [J. Anal. Math. 124, 261--295 (2014; Zbl 1351.37026)] for \(2\)-step nilsystems and work of \textit{A. M. Stepin} [Usp. Mat. Nauk 24, No. 5(149), 241--242 (1969; Zbl 0216.34501)] under a hypothesis on the homogeneous space, arguments due originally to \textit{W. Parry} [Topology 9, 217--224 (1970; Zbl 0176.20502)] are adapted here to prove that the space of square-integrable functions on an ergodic \(k\)-step nilsystem with \(k\geqslant2\) decomposes into a sum of a discrete spectrum subspace and a subspace with Lebesgue spectrum of infinite multiplicity. As a by-product an alternative proof of the \(k=2\) case is also given.