Vinogradov Circle Method

Theorem
$\newcommand{\A}{\mathcal A}$Let $\A$ be a subset of the non-negative integers.

For $\alpha \in \R$, let:
 * $e \left({\alpha}\right) := \exp \left({2 \pi i \alpha} \right)$

Let:


 * $\displaystyle T_N \left({s}\right) = \sum_{\substack {a \mathop \in \A \\ a \mathop \le N}} s^a$

be the truncated generating function for $\A$.

Let:
 * $V_N \left({\alpha}\right) := T_N \left({e \left({\alpha}\right)}\right)$

Let $r_{\A, \ell} \left({N}\right)$ be the number of solutions $\left({x_1, \dotsc, x_\ell}\right) \in \A^\ell$ to the equation:


 * $(1): \quad \displaystyle x_1 + \cdots + x_l = N$

Then:


 * $\displaystyle r_{\A, \ell} \left({N}\right) = \int_0^1 V_N \left({\alpha}\right)^\ell e \left({-N \alpha}\right) \, \mathrm d \alpha$

Proof
For $m \in \N$ let $r_{\A, \ell} \left({m; N}\right)$ be the number of solutions to $(1)$ with no $x_i$ exceeding $N$.

Then:
 * $\forall m \le N: r_{\A, \ell} \left({m; N}\right) = r_{\A, \ell} \left({m}\right)$

and:
 * $\forall m > \ell N: r_{\A, \ell} \left({m; N}\right) = 0$

Then we compute:


 * $\displaystyle T_N \left({s}\right)^\ell = \sum_{m \mathop = 0}^{\ell N} r_{\A, \ell} \left({m; N}\right) s^m$

and:


 * $(2): \quad \displaystyle V_N \left({\alpha}\right)^\ell = \sum_{m \mathop = 0}^{\ell N} r_{\A, \ell} \left({m; N}\right) e \left({\alpha m}\right)$

Now it follows from Exponentials Form Orthonormal Basis for $\mathcal L^2$ that:


 * $\displaystyle \int_0^1 e \left({\alpha m}\right) e \left({-\alpha n}\right) \, \mathrm d \alpha = \delta_{m n}$

where $\delta_{m n}$ is the Kronecker delta.

Therefore, we multiply $(2)$ by $e \left({-N \alpha}\right)$ and integrate:


 * $\displaystyle r_{\A, \ell} \left({N}\right) = r_{\A, \ell} \left({N; N}\right) = \int_0^1 \displaystyle V_N \left({\alpha}\right)^\ell e \left({-N \alpha}\right) \, \mathrm d \alpha $