Bromberg and Ulcigrai constructed piecewise smooth functions on the circle such that the set of for which the sum satisfies a temporal distributional limit theorem along the orbit of a.e. has Hausdorff dimension one. We show that the Lebesgue measure of this set is equal to zero.
[BU18] A temporal Central Limit Theorem for real-valued cocycles over rotations, Ann. Inst. Henri PoincarÃ©, Probab. Stat., Volume 54 (2018) no. 4, pp. 2304-2334 | Article | MR 2240787 | Zbl 1102.60005
[HW08] An introduction to the theory of numbers, Oxford University Press (2008), xxii+621 pages (Revised by D. R. Heath-Brown and J. H. Silverman, With a foreword by Andrew Wiles) | MR 1912816 | Zbl 0999.06006