Vinogradov's Theorem/Minor Arcs

Theorem
For sufficiently large odd integers $N$, and any $B > 10$:


 * $\displaystyle \int_{\mathfrak m}F(\alpha)^3 e(-\alpha N)\ d\alpha \ll \frac{N^2}{(\log N)^{B/2 - 5}}$

Lemma 1
For $\beta \in \R$, define $\|\beta\| = \min\{|n - \beta| : n \in \Z\}$.

Then for any $\alpha \in \R$,


 * $\displaystyle \left| \sum_{k = N_1}^{N_2} e(\alpha k) \right| \leq \min\left\{ N_2 - N_1, \frac1{2\|\alpha\|} \right\} $

Lemma 2
Let $a,q \in \Z$ such that:


 * $\displaystyle\left| \alpha - \frac aq \right| \leq \frac1{q^2},\quad 1 \leq q \leq N,\quad \operatorname{gcd}(a,q) = 1 \qquad (1)$

and let $m,n \in \Z$ be any integers. Then


 * $\displaystyle\sum_{k = 1}^m \min\left\{ \frac{mn}k,\frac1{\|\alpha k\|} \right\} \ll \left( m + \frac{mn}q + q \right) \log(2qm)$

Lemma 3
let $\alpha$ satisfy the condition $(1)$ of Lemma 3, let $x,y \in \N$ and let $\beta_k,\gamma_k$ be any complex numbers with $|\beta_k|, |\gamma_k| \leq 1$.

Then:


 * $\displaystyle \sum_{k = y}^{x/y} \sum_{\ell = y}^{x/k} \alpha_k \beta_\ell e(\alpha k\ell) \ll x^{1/2}(\log x)^2 \left( \frac xy + y + \frac xq + q \right)^{1/2}$