Almost sharp variational estimates for discrete truncated operators of Stein-Wainger type

We establish $r$-variational estimates for discrete truncated Stein-Wainger type operators on $\ell^p$ for $1<p<\infty$. Notably, these estimates are sharp and enhance the results obtained by Krause and Roos (J. Eur. Math. Soc. 2022, J. Funct. Anal. 2023), up to a logarithmic loss related to the scale. On the other hand, as $r$ approaches infinity, the consequences align with the estimates proved by Krause and Roos. Moreover, for the case of quadratic phases, we remove this logarithmic loss with respect to the scale in two and higher dimensions, at the cost of increasing $p$ slightly.