| The study of multiple recurrence averages was pioneered by Furstenberg in 1977, when he provided an alternative proof to Szemer'edi's theorem using ergodic theory, which states that a set of integer with a positive density contains an arbitrary long arithmetic progression. Since then, many convergence results of multiple recurrence averages have been obtained. Their norm convergence have been studied by Conze and Lesigne (in 1984), Host and Kra (2005), Ziegler (2006), Tao (2008), and the best result was obtained by Walsh (2012).;The results are much scarcer for pointwise convergence. In 1990, Bourgain answered Furstenberg's question by showing that some double recurrence averages converge pointwise. This deep result has not been generalized since then, while some partial results on the pointwise convergence of multiple recurrence averages are obtained by Derrien and Lesigne (1996), Assani (1998, 2005), and recently announced by Huang, Shao, and Ye (2014), and Donoso and Sun (2015). Also, Assani and Buczolich have shown that the pointwise convergence of double recurrence averages need not to hold when both functions are in L 1.;On the other hand, Brunel initiated the study of return times averages in his PhD thesis from 1966, where one concerns ergodic averages with weights that are generated randomly. In 2000, Assani showed that the sequence appearing in the multiple recurrence averages can be a good universal weight for multiple return times averages under some assumptions on the system.;We will show that one can extend Bourgain's double recurrence result in numerous ways. We will first show the Wiener-Wintner extension of the double recurrence theorem, which was a study initiated by Duncan in his doctoral dissertation from 2001. Furthermore, we will show a polynomial Wiener-Wintner result for the double recurrence theorem, extending the work of Lesigne (1990, 1993) and Frantzikinakis (2006). Secondly, from the angle of the Wiener-Wintner extension, we will show that the sequence appearing in the double recurrence averages can be a good universal weight for some nonconventional ergodic averages in norm, ultimately extending the work of Host and Kra, and Ziegler. |
