Vinogradov Circle Method

Theorem
Let $\AA$ be a subset of the non-negative integers.

For $\alpha \in \R$, let:
 * $\map e \alpha := \map \exp {2 \pi i \alpha}$

Let:


 * $\ds \map {T_N} s = \sum_{\substack {a \mathop \in \AA \\ a \mathop \le N} } s^a$

be the truncated generating function for $\AA$.

Let:
 * $\map {V_N} \alpha := \map {T_N} {\map e \alpha}$

Let $\map {r_{\AA, \ell} } N$ be the number of solutions $\tuple {x_1, \dotsc, x_\ell} \in \AA^\ell$ to the equation:


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

Then:


 * $\ds \map {r_{\AA, \ell} } N = \int_0^1 \map {V_N} \alpha^\ell \map e {-N \alpha} \rd \alpha$

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

Then:
 * $\forall m \le N: \map {r_{\AA, \ell} } {m; N} = \map {r_{\AA, \ell} } m$

and:
 * $\forall m > \ell N: \map {r_{\AA, \ell} } {m; N} = 0$

Then we compute:


 * $\ds \map {T_N} s^\ell = \sum_{m \mathop = 0}^{\ell N} \map {r_{\AA, \ell} } {m; N} s^m$

and:


 * $(2): \quad \ds \map {V_N} \alpha^\ell = \sum_{m \mathop = 0}^{\ell N} \map {r_{\AA, \ell} } {m; N} \map e {\alpha m}$

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


 * $\ds \int_0^1 \map e {\alpha m} \map e {-\alpha n} \rd \alpha = \delta_{m n}$

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

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


 * $\ds \map {r_{\AA, \ell} } N = \map {r_{\AA, \ell} } {N; N} = \int_0^1 \map {V_N} \alpha^\ell \map e {-N \alpha} \rd \alpha$