Introduction to additive combinatorics lecture 7.9 --- Basic Fourier transform properties

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Here I say a bit more about the discrete Fourier transform, including giving direct proofs of the basic properties that are used over and over again in arguments. As an example of how it relates to additive combinatorics, I show that the additive energy of a set can be expressed in a very clean way in terms of the Fourier transform of the characteristic function of that set.

0:00 Introduction
0:28 The discrete Fourier transform for Z_N and F_p^n
7:05 Statement and proof of the Parseval identity
15:12 Discrete delta functions
17:27 Statement and proof of the inversion formula
19:26 Definition of convolution
21:21 Statement and proof of the convolution law
25:39 Statement and proof of the dilation rule
32:00 Fourier transforms of characteristic functions
36:36 Fourier transforms and additive energy

Timothy Gowers

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