Polynomially moving ergodic averages

Abstract:

Given an increasing sequence of positive integers $\left \{ {{m_n}} \right \}$, a non-decreasing sequence of positive integers $\left \{ {{b_n}} \right \}$, and a measurable, measure-preserving ergodic transformation $\tau$ on a probability space $\left ( {\Omega ,\mathcal {F},\mu } \right )$, the a.s. convergence of the moving averages ${T_n}\left ( f \right ) = b_n^{ - 1}\sum \nolimits _{k = {m_n} + 1}^{{m_n} + {b_n}} {f\left ( {{\tau ^k}} \right )}$ is considered, for $f \in {L_p}\left ( \Omega \right )$. A counterexample is constructed in the case of polynomial-like $\left \{ {{m_n}} \right \}$.