Convergence of Conze-Lesigne averages (Q2716653)

A number of authors have studied limits of the form NEWLINE\[NEWLINE\lim_{n\to\infty} \tfrac{1}{N} \sum^{N-1}_{n=0} f_1(T^{a_1n}x)f_2(T^{a_2n}x)\dots f_l(T^{a_ln}x),NEWLINE\]NEWLINE for \(T\) an automorphism of a probability space \((X,{\mathcal B},\mu)\) and \(f_1, f_2,\dots, f_l\in L^\infty(X)\). The aim of this paper is to generalize a result of this type due to \textit{J.-P. Conze} and \textit{E. Lesigne} [Bull. Soc. Math. Fr. 112, 143-175 (1984; Zbl 0595.28018)] giving a new and simpler proof and giving a description of the limit. The authors prove:NEWLINENEWLINENEWLINELet \((X,{\mathcal B},\mu,T)\) be a measure preserving system, \(a_1,a_2,a_3\) be three distinct integers and \(f_1, f_2, f_3\) \(\in L^\infty(X,\mu)\). Then NEWLINE\[NEWLINE\lim_{n\to\infty} \tfrac 1N\sum^{N-1}_{n=0}\prod^3_{i=1}f_i(T^{a_in}x)NEWLINE\]NEWLINE exists in \(L^2(X,\mu)\).NEWLINENEWLINENEWLINEThe authors mention that they are unable to extend their proof to more than three terms.