2022
DOI: 10.4213/sm9356e
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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 $\max_x f(n,x,h)$ incr… Show more

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Cited by 3 publications
(5 citation statements)
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References 33 publications
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“…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.…”
Section: A Results By Kocherginsupporting
confidence: 54%
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“…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.…”
Section: A Results By Kocherginsupporting
confidence: 54%
“…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.…”
Section: Multidimensional Version Of Kochergin's Resultmentioning
confidence: 86%
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