(Uniform) Convergence of Twisted Ergodic Averages

Abstract

Let $T$ be an ergodic measure-preserving transformation on a non-atomic probability space $(X,Σ,μ)$. We prove uniform extensions of the Wiener-Wintner theorem in two settings: For averages involving weights coming from Hardy field functions, $p$: \[ \{\frac{1}{N} \sum_{n\leq N} e(p(n)) T^{n}f(x) \} \] and for "twisted" polynomial ergodic averages: \[ \{\frac{1}{N} \sum_{n\leq N} e(n θ) T^{P(n)}f(x) \} \] for certain classes of badly approximable $θ\in [0,1]$. We also give an elementary proof that the above twisted polynomial averages converge pointwise $μ$-a.e. for $f \in L^p(X), \ p >1,$ and arbitrary $θ\in [0,1]$.

31 pages, the referee's suggestions incorporated, references added, typos corrected. A uniform estimate of the ergodic averages with Hardy field weights by the corresponding Gowers-Host-Kra uniformity seminorms is added, see Theorem 2.11. To appear in Ergodic Theory Dynam. Systems