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The method of exponential sums is one of a few general methods enabling us to solve a wide range of miscellaneous problems from the theory of numbers and its applications. The strongest results have been obtained with the aid of this method. Therefore knowledge of the fundamentals of theory of exponential sums is necessary for studying modern number theory.
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An exponential sum estimate on small intervals
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 $y$ variabl …