Ergodic theorems along sequences and Hardy fields. (English) Zbl 0863.28011

Boshernitzan, Michael; Wierdl, Máté

Proc. Natl. Acad. Sci. USA 93, No. 16, 8205-8207 (1996).

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

Keywords:

Hardy fields; growth conditions; good averaging sequence; pointwise ergodic theorem; averages; mean convergence; pointwise convergence