Divergent square averages


In this paper we answer a question of J. Bourgain which was motivated by questions A. Bellow and H. Furstenberg. We show that the sequence $\{ n^{2}\}_{n=1}^{\infty}$ is $L^{1}$-universally bad. This implies that it is not true that given a dynamical system $(X ,\Sigma, \mu, T)$ and $f\in L^{1}(\mu)$, the ergodic means \[ \lim_{N\to \infty}\frac{1}N\sum _{n=1}^{N}f(T^{n^{2}}(x)) \] converge almost surely.