Title

Ergodic theorems along sequences and Hardy fields

Abstract

Let a(x) be a real function with a regular growth as x → (∞). [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)])(n=1)/((∞)) is a good averaging sequence in L2 for the pointwise ergodic theorem. A sequence (a(n)) of positive integers is a good averaging sequence in L2 for the pointwise ergodic theorem if in any dynamical system (Ω, Σ, m, T) for f ε L2(Ω) the averages 1/X Σ/n≤X f(T(a(n))ω) converge for almost every ω ε Ω. Our result implies that sequences like ([n(δ)]), where δ > 1 and not an integer, ([n log n]), and ([n2/log n]) are good averaging sequences for L2. 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 will 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) = log2 x), ([a(n)]) may be good for mean convergence without being good for pointwise convergence.

Publication Title

Proceedings of the National Academy of Sciences of the United States of America

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