Vinogradov mean value theorem：矩估计、Weyl sums 与 decoupling

旧博客原文

原题：Note on Vinogradov main theorem

1.Introduction

Question:
Vinogradov mean value
Let $k,s\in \mathbb N$ , $x\in R^k$ .

$J_{s,k}(N)=|\{(n_1,...,n_s,n_{s+1},...,n_{2s})|n_1^j+...+n_s^j=n_{s+1}^j+...+n_{2s}^j) \forall 1\leq j\leq k,1\leq n_i\leq N(1\leq i\leq s) \}|$
How to estimate $J_{s,k}(N)$ ?

We assume $f_{k}(x,N)=\sum_{1\leq n\leq N}e(nx_1+n^2x_2+...+n^kx_k)$ , then by following clear calculate:

$\int_{[0,1]^k}|f_k(x,N)|^{2s}dx_1...dx_k =\int_{[0,1]^k}|\sum_{1\leq n\leq N}e(nx_1+n^2x_2+...+n^kx_k)|^{2s}dx_1dx_2...dx_k &=\int_{[0,1]^k}\sum_{1\leq n_1,...,n_{2s}\leq N}e^{2\pi i[(n_1+...+n_{2s})x_1+...+(n_1^k+...+n_{2s}^k)x_k-(n_{s+1}+...+n_{2s})x_1-...-(n_{s+1}^k+...+n_{2s}^k)x_k]} &=|\{(n_1,...,n_{2s})| n_1^j+...+n_s^j=n_{s+1}^j+...+n_{2s}^j,\forall 1\leq j\leq k\}|$

we have:
$J_{s,k}(N)=\int_{[0,1]^k}|f_k(x,N)|^{2s}dx_1...dx_k$
main conjecture:
$\forall \epsilon >0$ ,we have:
$J_{s,k}(N)<<N^{\epsilon}(N^s+N^{2s-\frac{1}{2}k(k+1)})$

theorem(Bourgain-Demeter-Guth)
Main conjecture hold in general.

2.Application

We have following directly application:

1.Waring problem

2.Bound Weyl sums.\

3.Zero-free region for Riemann-zeta function.

3.Relate to the decoupling theorem

Now we discuss the decoupling theorem. This theorem describe the phenomenon when we are considering the “expension” operator $E_{[0,1]}(g)$ cut off $E_{[0,1]}(g)$ into a lot of small boxes $E_{J}(g)$ , then the $L_{d(d+1)})$ norms of the operator could be bounded very well, in fact it is near orthonagonal.

[B-D-G]
Let $d\geq 2$ ,$0<\delta\leq 1$. Then for each ball $B\subset R^d$ of radious at least $\delta^{-d}$ .
$||E_{[0,1]}g||_{L^{d(d+1)}(w_B)}<< \delta^{-\epsilon}(\sum_{J\subset [0,1],|J|=\delta}||E_Jg||^2_{L^{d(d+1)(w_B)}})^{\frac{1}{2}}$
( $J$ runs over a partition of $[0,1]$ in $\delta$ -intervals)

Discretized version:
Now we discuss the discretization of decoupling type result. We could establish a relationship between the decoupling theorem and Vinogradov mean theorem. look at the sum:
$\int_{[0,1]^k}|\sum_{1\leq n\leq N}e(nx_1+n^2x_2+...+n^kx_k)|^{2s}dx_1dx_2...dx_n$
This could be view as a $2s$ norm of a constant function $h=1$ , with a lebergue measure $d\sigma$ on curve $\Gamma=\{(t,t^2,...,t^d):0\leq t\leq 1\}$ . this curve $\Gamma$ could be view as a canonical curve with non-vanish guess curvature.
$||\widehat {hd\sigma}||_{2s}^{2s}=\int_{R^{k}}|\int_{\Gamma}h(t,t^2...,t^n)e(tx_1+...+t^kx_k)|^{2s}d\sigma$

this is very similar with the restriction theorem:

[restriction theorem]
let $\Gamma$ be $n-1$ dimension parabolic in $R^n$ , then guess curvature of $\Gamma$ is non-vanish. $\sigma$ is a natural induced lebergue measure on $\Gamma$ , we have, for suitable exponents $p,p'$ come from rescaling arument.
$||\widehat{gd\sigma}||_{p'}\lesssim ||g||_p$

So it seems like these are the same thing, but unfortunately they are not,there are two things distinct them:
1.the density is defferent, it is a discrete sum in:
$\int_{[0,1]^k}|\sum_{1\leq n\leq N}e(nx_1+n^2x_2+...+n^kx_k)|^{2s}dx_1dx_2...dx_n$
but a continue integral in:
$||\widehat {hd\sigma}||_{2s}^{2s}=\int_{R^k}|\int_{\Gamma}h(t,t^2...,t^n)e(tx_1+...+t^kx_k)|^{2s}d\sigma$
so we need to construct a rescaling way to make the discretization one coverage to the continue one.a suitable fexiable function seems like $(w_B,B_{r}(c_B)),w_B(x)=(1-\frac{|x-c_B|}{R})^{-100k}$
2.there

补充说明

以下是新整理的中文说明；上方旧博客原文保持不变。

Vinogradov mean value theorem 控制 Weyl sums 的高阶矩，是 Waring problem、指数和估计和 zeta 函数零点区域中的基础工具。Bourgain-Demeter-Guth 用 decoupling theorem 证明了主猜想。

Vinogradov mean value theorem：矩估计、Weyl sums 与 decoupling — Vinogradov mean value theorem 把 Weyl sums 的高阶矩与 moment curve decoupling 联系起来。

1. Mean value

设

$$S(\alpha)=\sum_{n\le N}e(\alpha_1n+\alpha_2n^2+\cdots+\alpha_kn^k).$$

Vinogradov mean value 研究

$$J_{s,k}(N)=\int_{[0,1]^k}|S(\alpha)|^{2s}\,d\alpha.$$

它也等于某个 Diophantine system 解的个数。

2. 主猜想

主猜想断言

$$J_{s,k}(N)\lesssim_\varepsilon N^\varepsilon\left(N^s+N^{2s-k(k+1)/2}\right).$$

两个项分别对应 diagonal solutions 和维数计数给出的主项。

3. 应用

这个估计直接用于 Waring problem，也给出 Weyl sums 的强上界。通过指数和控制，可以进一步进入 zeta 函数零点区域和等分布问题。

4. Decoupling 视角

考虑 moment curve

$$\gamma(t)=(t,t^2,\ldots,t^k).$$

decoupling theorem 描述 extension operator 在小区间分解后的 $L^p$ 几乎正交性。离散化后，它与 Vinogradov mean value theorem 精确相连。

5. 思想总结

Vinogradov mean value 把数论中的方程计数、调和分析中的 Fourier extension、以及几何中的曲率结构放到同一个问题里。这是现代解析数论和 decoupling 理论交汇的代表。