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The Van der Corput lemma states the following: Introduce the following oscillatory integral $$ I(a,b)=\int^{b}_{a}e^{ih(t)}dt. $$ Then

$(1)$ if $|h'(t)|\geq \lambda>0$ and $h'$ is monotonic, then we have the estimate $$ |I(a,b)|\leq C\lambda^{-1} $$ $(2)$ if $h\in C^k([a,b])$ and $|h^{(k)}(t)|\geq \lambda>0$, then we have the estimate $$ |I(a,b)|\leq C\lambda^{-\frac{1}{k}} $$ where the constant $C$ is independent of $a$ and $b$.

My question is, why is the Van der Corput lemma so important?

For example, we can bound the measure of a sublevel set. Why else is it important?

Thanks in advance for your comments and help.

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  • $\begingroup$ It's certainly used all over additive number theory, often in bounding the integrals involved in the Hardy-Littlewood Circle Method. I once went to a whole lecture course on such things, as a measure of the subject's depth. $\endgroup$ – Chappers Sep 13 '15 at 17:24
  • $\begingroup$ How is it used in the Hardy-Littlewood circle method? If possible, I am looking to compile a list of applications here. $\endgroup$ – user230715 Sep 13 '15 at 17:25
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The van der Corput lemma and associated inequalities are the technical heart of equidistribution theory in the interval and in the integers. It is via that connection that you find applications of van der Corput to the Hardy-Littlewood method (because bounding the contribution of the minor arcs is a statement about equidistribution of errors and therefore cancellation of errors).

The book Ten Lectures on the Interface between analytic number theory and harmonic analysis by Hugh Montgomery used this lemma over and over again, building on it (and four other fundamental inequalities). Even if you only care about harmonic analysis, this book demonstrates how to use van der Corput for harmonic analytic questions like bounding the size of fourier coefficients.

If you do not care about analytic number theory or harmonic analysis, you can look at Kuipers and Niederreiter Uniform Distribution of sequences where the van der Corput lemma is used to prove the main results on uniform distribution theory (parts of this exposition is also in Montgomery, though).

By the way, since you ask about list of applications, I have a three step roadmap to another answer of mine:

1) Van der Corput's lemma is a fundamental ingredient in uniform distribution theory.

2) Therefore, you should look for applications of uniform distribution theory.

3) I have given a detailed list in this post: Applications of equidistribution

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