One version of Weyl's inequality states that for any $\alpha\in\mathbb{R}$ and $(a, q) = 1$ such that $|\alpha - a/q|\le 1/q^2$, we have that $$\sum_{n\le X} e(n^k\alpha)\ll X^{1 + \varepsilon}(q^{-1} + X^{-1} + qX^{-k})^{2^{-k}}, e(\alpha) = e^{2\pi i\alpha}.$$
The proof uses Weyl differencing and proceeds by induction on $k$ and for the case $k = 1$, it uses the fact that $\sum_{n\le X} e(n\alpha) \ll \min(X, \lVert\alpha\rVert^{-1})$, where $\lVert\alpha\rVert$ is the smallest distance from an integer to $\alpha$. The proof of this fact very quickly follows by noting that this is juts a geometric series. This ceases to be the fact however if one considers instead the sum $\sum_{n\le X} e(n\alpha)w(n/X)$ for some smooth, quickly decaying, function $w$, though I think it is reasonable to expect the same bounds to still hold. Is it possible to obtain these bounds for this sum?
