Short Weyl sums and their applications

We shall study the behavior of short Weyl sums of the form
$$ T(\alpha ,x,y)=\sum_{x-y<m\leq x}e(\alpha m^n) $$
on major arcs and obtain an asymptotic formula for the number of representations of a sufficiently large positive integer $N$ as a sum of 33 fifth powers of positive integers $x_i$, that satisfy $ \left|x_i-\left(\dfrac{N}{33}\right)^{\frac 15}\right|\le H$, $H\ge N^{\frac 15-\frac{1}{340}+\varepsilon}$.
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