Rates of divergence of non-conventional ergodic averages


We first study the rate of growth of ergodic sums along a sequence (an) of times: SNf(x)= μn≤Nf(Tanx). We characterize the maximal rate of growth and identify a number of sequences such as an=2n, along which the maximal rate of growth is achieved. To point out though the general character of our techniques, we then turn to Khintchines strong uniform distribution conjecture that the averages (1/N) ∑n≤Nf(nx mod 1) converge pointwise to f for integrable functions f. We give a simple, intuitive counterexample and prove that, in fact, divergence occurs at the maximal rate. © 2009 Cambridge University Press.

Publication Title

Ergodic Theory and Dynamical Systems