Fractionally modulated discrete Carleson's Theorem and pointwise Ergodic Theorems along certain curves

Abstract

For $c\in (1,2)$ we consider the following operators \[ \mathcal{C}_{c}f(x) = \sup_{λ\in [-1/2,1/2)}\bigg| \sum_{n \neq 0}f(x-n) \frac{e^{2πiλ\lfloor |n|^{c} \rfloor}}{n}\bigg|\text{,}\quad \mathcal{C}^{\mathsf{sgn}}_{c}f(x) = \sup_{λ\in [-1/2,1/2)}\bigg| \sum_{n \neq 0}f(x-n) \frac{e^{2πiλ\mathsf{sign(n)} \lfloor |n|^{c} \rfloor}}{n}\bigg| \text{,} \] and prove that both extend boundedly on ${\ell }^p(\mathbb{Z})$, $p\in (1,\infty )$. The second main result is establishing almost everywhere pointwise convergence for the following ergodic averages \[ A_Nf(x)=\frac{1}{N}\sum_{n=1}^Nf(T^nS^{\lfloor n^c\rfloor}x)\text{,} \] where $T,S:X\to X$ are commuting measure-preserving transformations on a $\sigma$-finite measure space $(X,\mu )$, and $f\in L_{\mu }^p(X)$, $p\in (1,\infty )$. The point of departure for both proofs is the study of exponential sums with phases ${\xi }_2\lfloor |n^c|\rfloor +{\xi }_{1}n$ through the use of a simple variant of the circle method.

arXiv comment27 pages, no figures. Referee comments incorporated. Accepted for publication in Mathematische Zeitschrift