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

Let $μ(n)$ be the Möbius function, $e(z) = \exp(2πiz)$, $x$ real and $2\leq y \leq x$. This paper proves two sequences $(μ(n))$ and $(e(n^k α))$ are strongly orthogonal in short intervals. That is, if $k \geq 3$ being fixed and $y\geq x^{1-1/4+\varepsilon}$, then for any $A>0$, we have $\sum_{x< n \leq x+y} μ(n) e\left(n^k α\right) \ll y(\log y)^{-A}$ uniformly for $α\in \mathbb{R}$.

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