Hardy–Littlewood circle method

Definition

For real $\alpha$ write $e(\alpha) = e^{2\pi i \alpha}$. Let $\mathcal{S}$ be a finite set of positive integers, and let $s \ge 1$ and $N \ge 1$ be integers. Let $r(N)$ be the number of $s$-tuples $(x_1, \ldots, x_s) \in \mathcal{S}^s$ with $x_1 + \cdots + x_s = N$. Let $f(\alpha) = \sum_{x \in \mathcal{S}} e(x\alpha)$, an exponential sum. The circle method consists of the following steps.

1. Write $r(N) = \int_0^1 f(\alpha)^s\, e(-N\alpha)\, d\alpha$.
2. Choose a parameter $Q$ and split the circle $[0, 1)$ into major arcs $\mathfrak{M}$ and minor arcs $\mathfrak{m} = [0, 1) \setminus \mathfrak{M}$. The major arcs are the points near a rational $a/q$ with $q \le Q$ and $\gcd(a, q) = 1$.
3. On $\mathfrak{M}$, approximate $f$ by explicit functions to get a main term, usually the product of a singular series and a singular integral.
4. On $\mathfrak{m}$, bound $\abs{f}$ uniformly, for example by Weyl's inequality or Vinogradov's estimates, to show that $\int_{\mathfrak{m}}$ is smaller than the main term.