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Let $1<r<2$ be a real number. Let $4<p\le 6$. Consider the exponential sum estimate $$\int_0^{2\pi}\int_0^{N^{r-2}} \left|\sum_{n=1}^N e^{inx+in^2 y}\right|^p \, dy \, dx$$ Notice that the $y$ variable takes values in an interval of length much smaller than one. My question is, as $N\to \infty$, what is the sharp bound on the above exponential sum?

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