Multiple ergodic averages for tempered functions (Q1995536)

In 1977, \textit{H. Furstenberg} [J. Anal. Math. 31, 204--256 (1977; Zbl 0347.28016)] provided a purely ergodic theoretical proof of Szemere'di's theorem: every subset of natural numbers with positive upper density contains arbitrarily long arithmetic progressions, by studying the \(L^2\)-norm behavior as \(N\to \infty\) of its multiple ergodic averages. Following \textit{N. Frantzikinakis}'s approach [Trans. Am. Math. Soc. 367, No. 8, 5653--5692 (2015; Zbl 1351.37038); J. Anal. Math. 109, 353--395 (2009; Zbl 1186.37010); J. Anal. Math. 112, 79--135 (2010; Zbl 1211.37008)] on averages for Hardy field functions of different growth, the present author gives his contribution by studying the averages for tempered functions, a class which contains oscillating functions. The main result of the paper is the existence and the explicit expression of the \(L^2\)-norm limit of the aforementioned averages. The main ingredients are the use of the now classical PET induction, covering a more general case, namely a nice class of tempered functions and some equidistribution results on nilmanifolds.