(Uniform) Convergence of Twisted Ergodic Averages

Abstract

Let $T$ be an ergodic measure-preserving transformation on a non-atomic probability space $(X,\Sigma ,\mu )$. 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 $\theta \in [0,1]$. We also give an elementary proof that the above twisted polynomial averages converge pointwise $\mu$-a.e. for $f\in L^p(X),p>1,$ and arbitrary $\theta \in [0,1]$.

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

References in corpus (3)

• Polynomial Ergodic Averages Converge Rapidly: Variations on a Theorem of Bourgain
• Uniformity in the Wiener-Wintner theorem for nilsequences
• A random pointwise ergodic theorem with Hardy field weights

Cited by in corpus (4)

• Generic properties of extensions
• A random pointwise ergodic theorem with Hardy field weights
• Oscillation and jump inequalities for the polynomial ergodic averages along multi-dimensional subsets of primes
• Uniform vector-valued pointwise ergodic theorems for operators