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Rates of divergence of non-conventional ergodic averages

Abstract We first study the rate of growth of ergodic sums along a sequence (an) of times: SNf(x)=∑ n≤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 Khintchine’s 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.