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Square and rearrange

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Quick description

To control an integral \int_E f(x)\ d\mu(x) or a sum \sum_{n \in A} f(n), take its magnitude squared, expand it into a double integral \int_E \int_E f(x) \overline{f(y)}\ d\mu(x) d\mu(y) or a double sum \sum_{n \in A} \sum_{m \in A} f(n) \overline{f(m)}, and then rearrange, for instance by making the change of variables y=x+h or n=m+h.

This often has the effect of replacing a phase e^{2\pi i \phi(n)} or e^{2\pi i \phi(x)} in the original integrand by a "differentiated" phase such as e^{2\pi i (\phi(n+h)-\phi(n))} or e^{2\pi i (\phi(x+h)-\phi(x))}. Such differentiated phases are often more tractable to work with, especially if \phi had a "polynomial" nature to it.

Prerequisites

harmonic analysis, analytic number theory

Example 1

This is a classic example: to compute the integral = \int_{-\infty}^\infty e^{-\pi x^2}\ dx, square it to obtain

 A^2 = \int_{\R^2} e^{-\pi(x^2+y^2)}\ dx dy

then rearrange using polar coordinates to obtain

 A^2 = \int_0^2\pi \int_0^\infty e^{-\pi r^2} r\ dr d\theta.

The right-hand side can easily be evaluated to be 1, so the positive quantity = \int_{-\infty}^\infty e^{-\pi x^2}\ dx must also be 1.

Example 2

(Gauss sums)

Example 3

(Weyl sums)

Example 4

(The TT^* method, say to obtain Hormander's L^2 oscillatory integral estimate)

Example 5

(The large sieve)

General discussion

A variant of this trick is the van der Corput lemma for equidistribution.