Kusmin-Landau Inequality

Theorem

Let $I$ be the half-open interval $\hointl a b$.

Let $f: I \to R$ be continuously differentiable.

Let $f'$ be monotonic.

Let $\norm {f'} \ge \lambda$ on $I$ for some $\lambda \in \R_{>0}$, where $\norm {\, \cdot \,}$ denotes the distance to nearest integer.

Then:

$\ds \sum_{n \mathop \in I} e^{2 \pi i \map f n} = \map \OO {\frac 1 \lambda}$

where the big-$\OO$ estimate does not depend on $f$.

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Proof

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Source of Name

This entry was named for Rodion Osievich Kuzmin and Edmund Georg Hermann Landau.

Sources

• 1991: S.W. Graham and G. Kolesnik: Van der Corput's Method of Exponential Sums: Theorem $2.1$