Dynamical Systems Seminar

Bounded remainder sets for a dynamical system are sets for which the Birkhoff averages of return times differ from the expected values by at most a constant amount. These sets are rare and important objects which have been studied, especially in the context of Diophantine approximation, for over 100 years. In the last few years there have been a number of results which culminated in explicit constructions of bounded remainder sets for toral rotations in any dimension, of all possible allowable volumes. In these talks we are going to explain these results, and then explain how to generalize them to give explicit constructions of bounded remainder sets for rotations on the adelic torus. Our method of proof combines ideas from harmonic analysis on the adeles, dynamical systems, and the theory of mathematical quasicrystals.