Fractionally modulated discrete Carleson's Theorem and pointwise Ergodic Theorems along certain curves
Leonidas Daskalakis, Anastasios Fragkos
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 $T,S\colon X\to X$ are commuting measure-preserving transformations on a $σ$-finite measure space $(X,μ)$, and $f\in L_μ^p(X)$, $p\in(1,\infty)$. The point of departure for both proofs is the study of exponential sums with phases $ξ_2 \lfloor |n^c|\rfloor+ ξ_1n$ through the use of a simple variant of the circle method.