Spectral characterization of ergodic dynamical systems (Q2765431)

For any ergodic dynamical system \((X,\mu,T)\), the Wiener-Wintner ergodic theorem implies that NEWLINE\[NEWLINE a(N,T,f)=\sup_{\varepsilon>0}\|(1/N)\sum_{n=1}^{N} f(T^nx)e^{2\pi in\varepsilon}\|_2\to 0NEWLINE\]NEWLINE for any \(f\) in the ortho-complement of the closed linear span of the eigenfunctions of \(T\). While it is not possible to establish a uniform rate, it is shown here that there exists a function \(g\to\infty\) (depending on \(T\)) for which \(a(N,T,f)\leq C_f/g(N)\) for a dense set of such functions \(f\).NEWLINENEWLINEFor the entire collection see [Zbl 0972.00026].