The concept of ergodic averages lies at the heart of ergodic theory; it describes how certain quantities behave over long periods of time. A central question is whether, and in what sense, these averages converge. Classical results of Birkhoff and von Neumann from the 1930s established their convergence pointwise almost everywhere and in the $L^2$ norm.
The study of multiple ergodic averages was initiated in the 1970s by Furstenberg and his coauthors; their breakthrough work revealed a connection between ergodic theory and arithmetic combinatorics. The $L^2$ convergence of multiple ergodic averages was established by Tao in 2008, while the question of their pointwise almost everywhere convergence remains open to this day. In this talk, I will review the aforementioned results and then explain how techniques from harmonic analysis can be used to quantify the rate of $L^2$ convergence of these averages. Based on joint work with Floris van Doorn, Polona Durcik, Joris Roos and Christoph Thiele.