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? 



Exponential sums
There is the article [Primes in Special Intervals and Additive Problems with Such Numbers](http://link.springer.com/article/10.1023/A:1023270112701) of Maris Changa. He considered exponential sum with $n^l$ $(l>1)$ and estimated them using double sums
$$n^l\to n^l\left(1+\frac{xy}n\right)^l$$
and Vinogradov's mean value theorem. 


Will it help?


