I would like to estimate the following sum

$\sum_{N <n \leq 2N}e(vn^{l})$, $l \geq 1$ constant(not integer) and $v$ is a parameter(integer) that doesn't grow too fast(a small power of N).

The first idea may be to apply Weyl-Van der Corput inequality several times($[l]$ times) then invoke the theory of exponential pairs. But then, I get terms like $N^{1-1/2^{l}}$, which is not sufficient for my purposes. The other idea may be to apply Vinagradov's method. But this time a limitation sourced by h appears. Is there any other way to estimate the above sum?