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

For $c\in (1,2)$ we consider the following operators $${\mathcal{C}}_{c}f(x)=\underset{\lambda \in [-1\mathrm{/}2,1\mathrm{/}2)}{\sup }\mid \sum _{n\mathrm{\ne }0}f(x-n)\frac{e^{2\pi i\lambda \lfloor \mathrm{\mid }n{\mathrm{\mid }}^c\rfloor }}{n}\mid \text{,}{\mathcal{C}}_c^{\mathsf{s}\mathsf{g}\mathsf{n}}f(x)=\underset{\lambda \in [-1\mathrm{/}2,1\mathrm{/}2)}{\sup }\mid \sum _{n\mathrm{\ne }0}f(x-n)\frac{e^{2\pi i\lambda \mathsf{s}\mathsf{i}\mathsf{g}\mathsf{n}(\mathsf{n})\lfloor \mathrm{\mid }n{\mathrm{\mid }}^c\rfloor }}{n}\mid \text{,}$$ and prove that both extend boundedly on ${\mathrm{\ell }}^p(\mathbb{Z})$, $p\in (1,\mathrm{\infty })$. The second main result is establishing almost everywhere pointwise convergence for the following ergodic averages $$A_{N}f(x)=\frac{1}{N}\sum _{n=1}^{N}f(T^n{S}^{\lfloor n^c\rfloor }x)\text{,}$$ where $T,S\text{\hspace{0.17em}⁣}:X\to X$ are commuting measure-preserving transformations on a $\sigma$-finite measure space $(X,\mu )$, and $f\in L_{\mu }^p(X)$, $p\in (1,\mathrm{\infty })$. The point of departure for both proofs is the study of exponential sums with phases ${\xi }_2\lfloor \mathrm{\mid }{n}^c\mathrm{\mid }\rfloor +{\xi }_{1}n$ through the use of a simple variant of the circle method.