Behaviour of Birkhoff sums generated by rotations of the circle

Abstract

For continuous functions $f$ with zero mean on the circle we consider the Birkhoff sums $f(n,x,h)$ generated by the rotations by $2\pi h$, where $h$ is an irrational number. The main result asserts that the growth rate of the sequence $\max_x f(n,x,h)$ as $n \to \infty$ depends only on the uniform convergence to zero of the Birkhoff means $\frac{1}{n}f(n,x,h)$. Namely, we show that for any sequence $\sigma_k \to 0$ and any irrational $h$ there exists a function $f$ such that the sequence $\max_x f(n,x,h)$ increases faster than $n\sigma_n$. We also show that for any function $f$ that is not a trigonometric polynomial there exist irrational $h$ for which some subsequence $\max_x f(n_k,x,h)$ increases faster than the corresponding subsequence $n_k\sigma_{n_k}$. We present applications to weighted shift operators generated by irrational rotations and to their resolvents. Namely, we show that the resolvent of such an operator can increase arbitrarily fast in approaching the spectrum. Bibliography: 46 titles.