(Uniform) Convergence of Twisted Ergodic Averages

Let $T$ be an ergodic measure-preserving transformation on a non-atomic probability space $(X,\mathrm{\Sigma },\mu )$. We prove uniform extensions of the Wiener-Wintner theorem in two settings: For averages involving weights coming from Hardy field functions, $p$: $$\{\frac{1}{N}\sum _{n\le N}e(p(n))T^{n}f(x)\}$$ and for “twisted” polynomial ergodic averages: $$\{\frac{1}{N}\sum _{n\le N}e(n\theta )T^{P(n)}f(x)\}$$ 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 $f \in L^p(X), \ p >1,$andarbitrary$\theta \in [0,1]$.