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
Recommended Citation
Boshernitzan, M., & Wierdl, M. (1996). Ergodic theorems along sequences and Hardy fields. Proceedings of the National Academy of Sciences of the United States of America, 93 (16), 8205-8207. https://doi.org/10.1073/pnas.93.16.8205
