For \(c\in (1,2)\) we consider the following operators \(\begin{aligned} \mathcal {C}_{c}f(x)&{:=}\sup _{\lambda \in [-1/2,1/2)}\left| \sum _{n \ne 0}f(x-n) \frac{e^{2\pi i\lambda \lfloor |n|^{c} \rfloor }}{n}\right| {\text {,}}\\ \mathcal {C}^{\textsf{sgn}}_{c}f(x)&{:=}\sup _{\lambda \in [-1/2,1/2)}\left| \sum _{n \ne 0}f(x-n) \frac{e^{2\pi i\lambda \mathsf {sign(n)} \lfloor |n|^{c} \rfloor }}{n}\right| {\text {,}} \end{aligned}\) 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 \(T,S:X\rightarrow X\) 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 \(\xi _2 \lfloor |n^c|\rfloor + \xi _1n\) through the use of a simple variant of the circle method.
