1. Introduction
Question:
Vinogradov mean value
Let $k, s \in \mathbb{N}$, $x \in \mathbb{R}^k$.
$J_{s,k}(N) = |{(n_1, …, n_s, n_{s+1}, …, n_{2s}) | n_1^j + … + n_s^j = n_{s+1}^j + … + n_{2s}^j, \text{for all } 1 \leq j \leq k, 1 \leq n_i \leq N (1 \leq i \leq s)}|$
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:
$\int_{[0,1]^k} |f_k(x,N)|^{2s} dx_1…dx_k = \int_{[0,1]^k} |\sum_{1 \leq n \leq N} e(nx_1 + n^2x_2 + … + n^kx_k)|^{2s} dx_1dx_2…dx_k = \int_{[0,1]^k} \sum_{1 \leq n_1, …, n_{2s} \leq N} e^{2\pi i[(n_1 + … + n_{2s})x_1 + … + (n_1^k + … + n_{2s}^k)x_k – (n_{s+1} + … + n_{2s})x_1 – … – (n_{s+1}^k + … + n_{2s}^k)x_k]} = |{(n_1, …, n_{2s}) | n_1^j + … + n_s^j = n_{s+1}^j + … + n_{2s}^j, \text{for all } 1 \leq j \leq k}|$
We have:
$J_{s,k}(N) = \int_{[0,1]^k} |f_k(x,N)|^{2s} dx_1…dx_k$
Main conjecture:
For all $\epsilon > 0$, we have:
$J_{s,k}(N) \ll N^{\epsilon} (N^s + N^{2s – \frac{1}{2}k(k+1)})$
Theorem (Bourgain-Demeter-Guth):
Main conjecture holds in general.
2. Application
We have the following direct applications:
- Waring problem
- Bound Weyl sums.
- Zero-free region for Riemann-zeta function.
3. Related to the decoupling theorem
Now we discuss the decoupling theorem. This theorem describes the phenomenon when we are considering the “expansion” 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 nearly orthonogonal.
[B-D-G]
Let $d \geq 2$, $0 < \delta \leq 1$. Then for each ball $B \subset \mathbb{R}^d$ of radius at least $\delta^{-d}$.
$||E_{[0,1]}g||{L^{d(d+1)}(w_B)} \ll \delta^{-\epsilon} (\sum{J \subset [0,1], |J| = \delta} ||E_Jg||^2_{L^{d(d+1)}(w_B)})^{\frac{1}{2}}$
($J$ 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 viewed as a $2s$ norm of a constant function $h=1$, with a Lebesgue measure $d\sigma$ on the curve $\Gamma = {(t, t^2, …, t^d) : 0 \leq t \leq 1}$. This curve $\Gamma$ could be viewed as a canonical curve with non-vanishing Gauss curvature.
$||\widehat {hd\sigma}||{2s}^{2s} = \int{\mathbb{R}^{k}} |\int_{\Gamma} h(t, t^2…, t^n) e(tx_1 + … + t^kx_k)|^{2s} d\sigma$
This is very similar to the restriction theorem:
[Restriction theorem]
Let $\Gamma$ be $n-1$ dimension parabolic in $\mathbb{R}^n$, then the Gauss curvature of $\Gamma$ is non-vanishing. $\sigma$ is a natural induced Lebesgue measure on $\Gamma$, we have, for suitable exponents $p, p’$ come from rescaling argument.
$||\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:
- the density is different, 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 continuous integral in:
$||\widehat {hd\sigma}||{2s}^{2s} = \int{\mathbb{R}^k} |\int_{\Gamma} h(t, t^2…, t^n) e(tx_1 + … + t^kx_k)|^{2s} d\sigma$
so we need to construct a rescaling way to make the discretization one converge to the continuous one. A suitable flexible function seems like $(w_B, B_{r}(c_B)), w_B(x) = (1 – \frac{|x-c_B|}{R})^{-100k}$ - there