On the CLT for means under the rotation action. I

We propose a method allowing us to build, for various typical means generated by the action of any given irrational rotation of the circle, examples of $L^2$ functions satisfying the central limit theorem (CLT). We consider, for instance, nonlinear means, and means along the sequence of squares. In the latter case, the circle method of Hardy and Littlewood is used. We also give an example of continuous Gaussian random Fourier series with sample paths satisfying both the CLT and the almost sure CLT.