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Ergodic theorems along sequences and Hardy fields. (English) Zbl 0863.28011
Summary: Let $$a(x)$$ be a real function with a regular growth as $$x\to\infty$$. [The precise technical assumption is that $$a(x)$$ belongs to a Hardy field.] We establish sufficient growth conditions on $$a(x)$$ so that the sequence $$([a(n)])^\infty_{n=1}$$ is a good averaging sequence in $$L^2$$ for the pointwise ergodic theorem. A sequence $$(a_n)$$ of positive integers is a good averaging sequence in $$L^2$$ for the pointwise ergodic theorem if in any dynamical system $$(\Omega,\Sigma,m,T)$$ for $$f\in L^2(\Omega)$$ the averages ${1\over X} \sum_{n\leq X} f(T^{a_n}\omega)$ converge for almost every $$\omega\in\Omega$$. Our result implies that sequences like $$([n^\delta])$$, where $$\delta>1$$ and not an integer, $$([n\log n])$$ and $$([n^2/\log n])$$ are good averaging sequences for $$L^2$$. In fact, all the sequences we examine will turn out to be good averaging for $$L^p$$, $$p>1$$; and even for $$L\log L$$.
We also establish necessary and sufficient growth conditions on $$a(x)$$ so that the sequence $$([a(n)])$$ is good averaging for mean convergence. Note that for some $$a(x)$$ (e.g., $$a(x)=\log^2x)$$, $$([a(n)])$$ may be good for mean convergence without being good for pointwise convergence.

##### MSC:
 28D05 Measure-preserving transformations 26A12 Rate of growth of functions, orders of infinity, slowly varying functions
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