Local Fourier uniformity of higher divisor functions on average

Let ${\tau }_k$ be the $k$-fold divisor function. By constructing an approximant of ${\tau }_k$, denoted as ${\tau }_k^{\ast }$, which is a normalized truncation of the $k$-fold divisor function, we prove that when $\exp (C{\log }^{1\mathrm{/}2}X(\log \log X{)}^{1\mathrm{/}2})\le H\le X$ and $C>0$ is sufficiently large, the following estimate holds for almost all $x\in [X,2X]$: $$\mid \sum _{x<n\le x+H}({\tau }_k(n)-{\tau }_k^{\ast }(n))e({\alpha }_d{n}^d+\cdots +{\alpha }_{1}n)\mid =o(H{\log }^{k-1}X),$$ where ${\alpha }_1,\dots ,{\alpha }_d\in \mathbb{R}$ are arbitrary frequencies.