Suppose \(a\)is an irrational number. Weyl proved that the frequency of visits of \(na\;{\rm mod}\; 1\) to a subinterval of [0,1] tends to the length of the subinterval ("uniform distribution"). It has long been known that the error term in this limit theorem can exhibit strong bias (“irregularity in uniform distribution"). This bias depends on the fine number theoretic properties of \(a\). For example, the square roots of two and three do not behave in the same way (!)

I will explain how infinite ergodic theory can shed light on this phenomenon, and 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.