ATS Theorem - Van Der Corput Lemma

Van Der Corput Lemma

Let be a real differentiable function in the interval moreover, inside of this interval, its derivative is a monotonic and a sign-preserving function, and for the constant such that satisfies the inequality Then

$\sum_{a<k\le b} e^{2\pi i f(k)} = \int_a^be^{2\pi i f(x)}dx + \theta\left(3 + \frac{2\delta}{1-\delta}\right),$

where

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