Randomness of Birkhoff sums
Bence Borda (Sussex)
The rotation of the circle group R/Z by a fixed irrational angle and its higher dimensional analogue on the d-dimensional torus are one of the most classical examples of a dynamical system. Even though these systems are not chaotic, Birkhoff sums associated to them exhibit remarkable randomness. We survey some old and very recent results in the area, and discuss various approaches. In particular, we describe how spectral properties of the Gauss map and its generalisations related to continued fraction expansions lead to probabilistic limit theorems.
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