= Van der Corput sum-integral lemma
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{title2=$\sum_{a<n\le b}e(f(n))=\int_a^be(f(x))\,dx+O((1-\delta)^{-1})$}
For a real $C^1$ phase with continuous <monotone> derivative and $|f'|\le\delta<1$, the sum-integral discrepancy has the displayed uniform bound. Periodization and the <Dirichlet-Jordan convergence theorem> express it as symmetric nonzero <Fourier modes>. <Integration by parts> gives endpoint terms and reciprocal-derivative variations. Monotonicity bounds their total variation by $\sum_{h\ne0}2\delta/(h^2-\delta^2)=O((1-\delta)^{-1})$. Pairing the leading $1/h$ endpoint terms reduces them to the uniformly bounded sine series. The exclusion of integer nonzero frequencies is essential.
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