  Let $\mu(n)$ be the M\"obius function, $e(z) = \exp(2\pi iz)$, $x$ real and
$2\leq y \leq x$. This paper proves two sequences $(\mu(n))$ and $(e(n^k
\alpha))$ 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 \[
  \sum_{x< n \leq x+y} \mu(n) e\left(n^k \alpha \right) \ll y(\log y)^{-A} \]
uniformly for $\alpha \in \mathbb{R}$.

Number Theory