The classical Birkhoff ergodic theorem states that in any measure preserving system, the 'time-averages' of a function along orbits of the measure preserving transformation converge pointwise. Bourgain extended this result to certain double ergodic averages. This talk is about a quantitative version of Bourgain's result: How many jumps larger than $\epsilon$ can the sequence of double averages have, before it eventually converges? This is joint work with Polona Durcik.