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.
Ergodic Theory and Dynamical Systems
Quas, A., & Wierdl, M. (2010). Rates of divergence of non-conventional ergodic averages. Ergodic Theory and Dynamical Systems, 30 (1), 233-262. https://doi.org/10.1017/S0143385709000054