On central limit theorems,modulus of continuity and diophantine type for irrational rotations

LetR _α be a rotation on the circle by an irrational angle α. LetB(t) be a Brownian motion (for instance). Then (Lacey (1990), (1991)) there is anf ∈L ² so that $$m^{ - 1/2} (f + ... + f o R_\alpha ^{m1] - 1} )\mathop \Rightarrow \limits^d B(t)$$ In this note, we show thatf can be taken to be continuous, and give a sharp estimate on the modulus of continuity off, in terms of number-theoretic properties of α. The same result is given for self-similar processes other than the Brownian motion.