Vinogradov Circle Method
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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 $\AA$.
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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:
- $\ds \map {r_{\AA, \ell} } N = \map {r_{\AA, \ell} } {N; N} = \int_0^1 \map {V_N} \alpha^\ell \map e {-N \alpha} \rd \alpha$
$\blacksquare$
Source of Name
This entry was named for Ivan Matveevich Vinogradov.