Polynomially moving ergodic averages

Abstract

Given an increasing sequence of positive integers { m n } \left \{ {{m_n}} \right \} , a non-decreasing sequence of positive integers { b n } \left \{ {{b_n}} \right \} , and a measurable, measure-preserving ergodic transformation τ \tau on a probability space ( Ω , F , μ ) \left ( {\Omega ,\mathcal {F},\mu } \right ) , the a.s. convergence of the moving averages T n ( f ) = b n − 1 ∑ k = m n + 1 m n + b n f ( τ k ) {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 ∈ L p ( Ω ) f \in {L_p}\left ( \Omega \right ) . A counterexample is constructed in the case of polynomial-like { m n } \left \{ {{m_n}} \right \} .