In Chapter 1 we have already seen the power of the probabilistic method in additive combinatorics, in which one understands the additive structure of a random object by means of computing various averages or moments of that object. In this chapter we develop an equally powerful tool, that of Fourier analysis. This is another way of computing averages and moments of additively structured objects; it is similar to the probabilistic method but with an important new ingredient, namely that the quantities being averaged are now “twisted” or “modulated” by some very special complex-valued phase functions known as characters. This gives rise to the concept of a Fourier coefficient of a set or function, which measures the bias that object has with respect to a certain character. These coefficients serve two major purposes in this theory. Firstly, one can exploit the orthogonality between different characters to obtain non-trivial bounds on these coefficients; this orthogonality plays a role somewhat similar to the role of independence in probability theory. Secondly, Fourier coefficients are very good at controlling the operation of convolution, which is the analog of the sum set operation, but for functions instead of sets. Because of this, the Fourier transform is ideal for studying certain arithmetic quantities, most notably the additive energy introduced in Definition 2.8.
Using Fourier analysis, one can essentially divide additive sets A into two classes. At one extreme are the pseudo-random sets, whose Fourier transform is very small (except at 0); we shall introduce the linear bias ∥A∥u and the Λ (p) constants to measure this pseudo-randomness.
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