On the Growth of Birkhoff Sums over a Rotation of the Circle

Even Poincaré had constructed an example which implies the existence of an irrational rotation of the circle and a function continuous on it with zero mean for which the Birkhoff sums at some points tend to infinity as the number of iterations grows. The strict ergodicity in this case is a natural constraint on the growth rate of Birkhoff sums: the sequence of Birkhoff means uniformly tends to zero on the circle. The paper shows that any prescribed admissible rate of growth of Birkhoff sums within the ergodic theorem can be realized, and the set of points at which the sums grow with a given speed is massive: it has the Hausdorff dimension one.