Better than square-root cancellation in Piatetski-Shapiro sequences

Abstract

In this paper, we investigate whether the better than square-root cancellation phenomenon exists for $\sum^{}_{n\leq X,n\in \mathcal{A}}f(n)$ when $\mathcal{A}$ is a Piatetski-Shapiro sequence and $f(n)$ is a Steinhaus or Rademacher random multiplicative function. Harper's remarkable breakthrough (2019) showed that better than square-root cancellation phenomenon happens when $\mathcal{A}$ takes natural integers set $\mathbb{N}$. Then Max Wenqiang Xu (2023) proved the conclusion also holds if $\mathcal{A}$ consists of $\mathcal{R}$-rough numbers. The similar result can be obtained for $y$-smooth numbers according to Hardy and Xu's recent paper(2026). Our result provides another positive example about the existence of better than square-root cancellation phenomenon when $\mathcal{A}$ is not a set with multiplicative energy as small as $(2+o(1))|\mathcal{A}|^2$. Furthermore, inspired by Harper's work (2023), we also prove the typical size of character sums over Piatetski-Shapiro sequences is $o(\sqrt{|\mathcal{N}_c(x)|})$. Based on this, the character sums over Piatetski-Shapiro sequences can reach Weil's bound for almost all characters modulo a prime $p$.