A recurrence lemma and its applications and extensions

I will discuss a lemma which is usually attributed to Atkinson, is very useful and apparently has been rediscovered multiple times. The lemma says that if (X, B, mu, T) is an ergodic system and f is in L^1(mu) with mean zero, then the Birkhoff sums of f do not diverge almost surely. Moreover the sums switch signs in a wide sense, infinitely many times. I will give the proof and discuss some applications of this result. Then I will discuss higher dimensional analogues, where the situation is significantly more complicated.