Note on Vinogradov main theorem



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

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

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

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

Main conjecture hold in general.


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 $latex E_{[0,1]}(g)$ cut off $E_{[0,1]}(g)$ into a lot of small boxes $latex E_{J}(g)$, then the $L_{d(d+1)})$ norms of the operator could be bounded very well, in fact it is near orthonagonal.

Let $latex d\geq 2$,$0<\delta\leq 1$. Then for each ball $latex B\subset R^d$ of radious at least $latex \delta^{-d}$.
$latex ||E_{[0,1]}g||_{L^{d(d+1)}(w_B)}<< \delta^{-\epsilon}(\sum_{J\subset [0,1],|J|=\delta}||E_Jg||^2_{L^{d(d+1)(w_B)}})^{\frac{1}{2}}$
($latex J$ runs over a partition of $latex [0,1]$ in $latex \delta$-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:
$latex \int_{[0,1]^k}|\sum_{1\leq n\leq N}e(nx_1+n^2x_2+…+n^kx_k)|^{2s}dx_1dx_2…dx_n$
This could be view as a $latex 2s$ norm of a constant function $latex h=1$, with a lebergue measure $latex d\sigma$ on curve $latex \Gamma=\{(t,t^2,…,t^d):0\leq t\leq 1\}$. this curve $latex \Gamma$ could be view as a canonical curve with non-vanish guess curvature.
$latex ||\widehat {hd\sigma}||_{2s}^{2s}=\int_{R^{k}}|\int_{\Gamma}h(t,t^2…,t^n)e(tx_1+…+t^kx_k)|^{2s}d\sigma$

this is very similar with the restriction theorem:

[restriction theorem]
let $latex \Gamma$ be $n-1$ dimension parabolic in $latex R^n$, then guess curvature of $latex \Gamma$ is non-vanish.$latex \sigma$ is a natural induced lebergue measure on $latex \Gamma$, we have, for suitable exponents $latex p,p’$ come from rescaling arument.
$latex ||\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:
$latex \int_{[0,1]^k}|\sum_{1\leq n\leq N}e(nx_1+n^2x_2+…+n^kx_k)|^{2s}dx_1dx_2…dx_n$
but a continue integral in:
$latex ||\widehat {hd\sigma}||_{2s}^{2s}=\int_{R^k}|\int_{\Gamma}h(t,t^2…,t^n)e(tx_1+…+t^kx_k)|^{2s}d\sigma$
so we need to construct a rescaling way to make the discretization one coverage to the continue one.a suitable fexiable function seems like $latex (w_B,B_{r}(c_B)),w_B(x)=(1-\frac{|x-c_B|}{R})^{-100k}$