Pointwise convergence of averages along cubes

Let $(X,\mathcal{B},\mu, T)$ be a measure preserving system. We prove the pointwise convergence of the averages $$\frac{1}{N^2}\sum_{n,m= 0}^{N-1} f_1(T^nx)f_2(T^mx)f_3(T^{n+m}x)$$ and of similar averages with seven bounded functions.Comment: 18 pages, latex, We have replaced Lemma 2 with a new one. We also have added a referenc

Extension of Wiener-Wintner double recurrence theorem to polynomials

We extend the authors' previous work on Wiener-Wintner double recurrence theorem to the case of polynomials.Comment: A few typos corrected and more details given in the proof of Theorem 3.