Local Fourier uniformity of higher divisor functions on average

Abstract

Let $\tau_k$ be the $k$-fold divisor function. By constructing an approximant of $\tau_k$, denoted as $\tau_k^*$, which is a normalized truncation of the $k$-fold divisor function, we prove that when $\exp\left(C\log^{1/2}X(\log\log X)^{1/2}\right)\leq H\leq X$ and $C>0$ is sufficiently large, the following estimate holds for almost all $x\in[X,2X]$: \[ \Big|\sum_{x<n\leq x+H}(\tau_k(n)-\tau_k^*(n)) e(\alpha_dn^d+\cdots+\alpha_1n)\Big|=o(H\log^{k-1}X), \] where $\alpha_1, \dots, \alpha_d\in \mathbb{R}$ are arbitrary frequencies.