Pointwise characteristic factors for Wiener-Wintner double recurrence theorem (Q2813940)

In the paper, the authors extend Bourgain's double recurrence result to the Wiener-Wintner averages.NEWLINENEWLINEThe main result is the following: Let \((X,\mathcal{F},\mu,T)\) be a standard ergodic dynamical system, and \(f_1, f_2 \in L^2 (X)\). Let NEWLINE\[NEWLINEW_N (f_1,f_2,x,t)=\frac{1}{N} \sum_{n=0}^{N-1} f_1 (T^an x) f_2 (T^bn x)e^{2\pi int}.NEWLINE\]NEWLINE (1) Double uniform Wiener-Wintner Theorem. If either \(f_1\) or \(f_2\) belongs to \(\mathcal{Z}_2^{\perp}\), where \(\mathcal{Z}_2\) is the Conze-Lesigne factor, then there exists a set of full measure \(X_{f_1 \otimes f_2}\) such that for all \(x \in X_{f_1 \otimes f_2}\), NEWLINE\[NEWLINE\limsup_{N\rightarrow\infty} \sup_{t\in\mathbb{R}} |W_N (f_1,f_2,x,t)|=0.NEWLINE\]NEWLINE (2) General convergence. If \(f_1, f_2 \in \mathcal{Z}_2\), then for \(\mu\)-a.e. \(x\in X\), \(W_N (f_1,f_2,x,t)\) converges for all \(t\in\mathbb{R}\).