Linear Sequences and Weighted Ergodic Theorems

Abstract

We present a simple way to produce good weights for several types of ergodic theorem including the Wiener-Wintner type multiple return time theorem and the multiple polynomial ergodic theorem. These weights are deterministic and come from orbits of certain bounded linear operators on Banach spaces. This extends the known results for nilsequences and return time sequences of the form $(g(S^{n}y))$ for a measure preserving system $(Y,S)$ and $g\in L^{\infty }(Y)$, avoiding in the latter case the problem of finding the full measure set of appropriate points $y$.