Dolgopyat, Dmitry; Sarig, Omri
No temporal distributional limit theorem for a.e. irrational translation
Annales Henri Lebesgue, Volume 1  (2018), p. 127-148

### Abstract

Bromberg and Ulcigrai constructed piecewise smooth functions on the circle such that the set of $\alpha$ for which the sum ${\sum }_{k=0}^{n-1}f\left(x+k\alpha \phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}1\right)$ satisfies a temporal distributional limit theorem along the orbit of a.e. $x$ has Hausdorff dimension one. We show that the Lebesgue measure of this set is equal to zero.

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