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.

Distribution of Kronecker sequences

“…A similar result without the condition on thickness of the set D (just an existence result) was published a little bit earlier in [1]. In the present paper we are mostly interested just in existence-type results.…”

“…We are interested here only in the behaviour of a particular sum with x x x = 0 0 0 for the rotation of torus T n . Our proof follows the main argument from [1,15] and is related to the best Diophantine approximations.…”

“…In papers [1,15] function f is constructed as a sum of a series which converges uniformly and which elements are piecewise linear functions.…”

“…In the case when A ∈ − 1 λ , −1 is close to − 1 λ the series (1) is unbounded and oscillates. Poincaré obtained series (1) as a solution of a certain simple differential equation, in particular, it can be represented as an integral…”

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This short and simple communication is motivated by recent papers [1,5,15] as well as by the discussion on Poincaré's and Bohl's results in [35], Chapter 8. We give a brief analysis of an example constructed by Poincaré and its recent generalisations.

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