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Demeter in his book "Fourier Restriction, Decoupling, and Applications" (P287) used the following estimate:

\begin{equation} \sup_{0\leq n\leq q}\bigg|\sum_{m=0}^n e^{2\pi i\frac{a}{q}m^2}\bigg|\leq Cq^{\frac{1}{2}}, \end{equation} where $a,q\in \mathbb{N}_+$ satisfy $(a,q)=1$ and $2\nmid q$.

The case $n=q$ can be calculated directly to be $q^{\frac{1}{2}}$. For general $n$, the usual Weyl's method will lead to an extra term $\log n$. Could you please explain how to get this estimate?

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    $\begingroup$ Is this what you want? $\endgroup$
    – mathworker21
    Commented Oct 25, 2022 at 18:33
  • $\begingroup$ This helps a lot. Thank you. $\endgroup$
    – Dapao Zhang
    Commented Oct 26, 2022 at 2:49

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The main idea is to apply Poisson summation formula. It allows to replace $a/q$ by $\lfloor q/a\rfloor$ and then repeat the procedure using continued fraction expansion of $a/q$. This idea belongs to Hardy and Littlewood, but afterwards it was rediscovered many times. You can find references and explicite estimates in the article On Incomplete Gaussian Sums by M. A. Korolev.

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    $\begingroup$ Thank you! Does there exist a relatively simple proof? The sharp constant is not important to me. $\endgroup$
    – Dapao Zhang
    Commented Oct 26, 2022 at 2:51
  • $\begingroup$ @DapaoZhang Unfortunately I don't know where you can find the simple proof. It is always necessary to deal with the error term in Poisson summation formula. $\endgroup$
    – Alexey Ustinov
    Commented Oct 26, 2022 at 7:21

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