(AD) 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 have culminated in explicit constructions of bounded remainder sets for toral rotations in any dimension, of all possible allowable volumes. In this talk, we are going to give a survey of these results, the recent constructions of bounded remainder sets for rotations on the adelic torus by Alan Haynes, Joanna Furno and Henna Koivusalo and finally give a brief description of the construction of bounded remainder sets for rotations on the adelic torus in any dimension. Our results combine ideas from harmonic analysis, dynamical systems, and the theory of mathematical quasicrystals. This is joint work with Alan Haynes and Joanna Furno.