Exponential Sums by Irrationality Exponent

In this article, we give an asymptotic bound for the exponential sum of the M\"obius function $\sum_{n \le x} \mu(n) e(\alpha n)$ for a fixed irrational number $\alpha\in\mathbb{R}$. This exponential sum was originally studied by Davenport and he obtained an asymptotic bound of $x(\log x)^{-A}$ for any $A\ge0$. Our bound depends on the irrationality exponent $\eta$ of $\alpha$. If $\eta \le 5/2$, we obtain a bound of $x^{4/5 + \varepsilon}$ and, when $\eta \ge 5/2$, our bound is $x^{(2\eta-1)/2\eta + \varepsilon}$. This result extends a result of Murty and Sankaranarayanan, who obtained the same bound in the case $\eta = 2$.