Strong orthogonality between the Mobius function and nonlinear exponential functions in short intervals

Let $\mu (n)$ be the M"obius function, $e(z)=\exp (2\pi iz)$, $x$ real and $2\le y\le x$. This paper proves two sequences $(\mu (n))$ and $(e(n^k \alpha))$arestronglyorthogonalinshortintervals\mathrm{.}Thatis,if$k \geq 3$beingfixedand$y\geq x^{1-1/4+\varepsilon}$,thenforany$A>0$, we have $$\sum _{x<n\le x+y}\mu (n)e\left(n^k\alpha \right)\ll y(\log y{)}^{-A}$$ uniformly for $\alpha \in \mathbb{R}$.