Convergence of diagonal ergodic averages
Convergence of diagonal ergodic averages
Tao has recently proved that if $T_1,...,T_l$ are commuting, invertible, measure-preserving transformations on a dynamical system then for any $L^\infty$ functions $f_1,...,f_l$, the average $\frac{1}{N}\sum_{n=0}^{N-1}\prod_{i\leq l}f_i\circ T^n_i$ converges in the $L^2$ norm. Tao's proof is unusual in that it translates the problem into a more complicated statement about the combinatorics …