(Uniform) convergence of twisted ergodic averages

Abstract

Let $T$ be an ergodic measure-preserving transformation on a non-atomic probability space $(X,\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D707})$ . We prove uniform extensions of the Wiener–Wintner theorem in two settings: for averages involving weights coming from Hardy field functions $p$ , $$\begin{eqnarray}\displaystyle \bigg\{\frac{1}{N}\mathop{\sum }_{n\leq N}e(p(n))T^{n}f(x)\bigg\}; & & \displaystyle \nonumber\end{eqnarray}$$ and for ‘twisted’ polynomial ergodic averages, $$\begin{eqnarray}\displaystyle \bigg\{\frac{1}{N}\mathop{\sum }_{n\leq N}e(n\unicode[STIX]{x1D703})T^{P(n)}f(x)\bigg\} & & \displaystyle \nonumber\end{eqnarray}$$ for certain classes of badly approximable $\unicode[STIX]{x1D703}\in [0,1]$ . We also give an elementary proof that the above twisted polynomial averages converge pointwise $\unicode[STIX]{x1D707}$ -almost everywhere for $f\in L^{p}(X),p>1,$ and arbitrary $\unicode[STIX]{x1D703}\in [0,1]$ .

Locations

• Ergodic Theory and Dynamical Systems - View
• arXiv (Cornell University) - View - PDF
• MPG.PuRe (Max Planck Society) - View - PDF
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