Irregularities in Uniform Distribution

Suppose a is an irrational number. Weyl proved that na mod 1 is uniformly distributed on the unit interval. i.e., the frequency of visits of na mod 1 to a subinterval of [0,1] tends to the length of the subinterval. It has long been known that the error term in this limit theorem can exhibit strong bias, reflecting an “irregularity” in uniform distribution in the higher-order term. This bias depends on the fine number theoretic properties of a. For example, the square roots of two and three do not behave the same way (!) I will describe some recent joint work with Dmitry Dolgopyat on the equidistribution of the error term. There are amusing connections to the geometry of translation surfaces with infinite genus, and to local limit theorems of inhomogeneous Markov chains