Coboundaries in L ∞₀

Let T be an ergodic automorphism of a probability space, f a bounded measurable function, S_n(f) = ∑_k=0^n-1 f ○ T^k. It is shown that the property that the probabilities μ( S_n(f) > n) are of order n^-p roughly corresponds to the existence of an approximation in L ∞ of f by functions (coboundaries) g - g ○ T, g ∈ L^p. Similarly, the probabilities μ( S_n(f) > n) are exponentially small iff f can be approximated by coboundaries g - g ○ T where g have finite exponential moments.