Marius Neubert (Leipzig)

Abstract:

A natural generalization of the classical problem of finding good weights for the pointwise ergodic theorem is the question of uniformity. For instance, Bourgain's Return Time Theorem is probabilistic in nature. It shows that for any measure preserving system and any of its essentially bounded functions, almost every point's orbit under the function is a good weight. However, there is no nice condition to characterize these "nice" points. 
The uniform version of this question has not been answered yet. The big question is: Which restrictions do we have to make on the system? We discuss several possibilities and show why they are all failing. In the end, we give a partial positive answer when we have uniformity -- by diving into the class of LR and substitutions systems and also prove a Uniform Return times theorem for the Thue-Morse system.
This is joint ongoing work with Philipp Gohlke and Felix Pogorzelski.