Linear sequences and weighted ergodic theorems

TL;DR

This paper introduces a method to generate deterministic weights for various ergodic theorems using orbits of bounded linear operators on Banach spaces, extending previous results on nilsequences and return time sequences.

Contribution

It provides a new approach to produce weights for ergodic theorems from linear operator orbits, broadening the scope beyond nilsequences and specific return time sequences.

Findings

Weights are deterministic and derived from linear operator orbits.

Extends known results to more general weights in ergodic theorems.

Avoids the need for full measure sets of points in certain cases.

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^ny)) 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.