论文标题
二次vinogradov平均值定理的直径估计值
Diameter free estimates for the quadratic Vinogradov mean value theorem
论文作者
论文摘要
令$ s \ geq 3 $为天然数字，让$ψ（x）$为多项式，具有真实系数和度量$ d \ geq 2 $，然后让$ a $是一些大的，非空的，有限的实际数字子集。我们使用$ e_ {s，2}（a）$来表示方程系统的解决方案\ [\ sum_ {i = 1}^{s}^{s}（ψ（x_i） - ψ（x__ {x_ {i+s}）= \ sum________i = 1}}^{i = 1}^{i = 1}^{i = x_____i = i+s} \在$中，每$ 1 \ leq i \ leq 2s $。我们的主要结果表明，\ [e_ {s，2}（a）\ ll_ {d，s} | a |^{2s -3 +η_{s}}，\]，其中$η_3= 1/2 $，$η_3= 1/2 $，$η_{s} =（s} =（1/4-1/4-1/4-1/7246）这种风味的唯一其他先前已知的结果是由于Bourgain和Demeter所致，他们表明，当$ψ（x）= x^2 $和$ s = 3 $时，我们有\ [e_ {3,2}（a）\ll_ε| a |^{3 + 1/2 +ε}，每个$^ + 1/2 + 1/2 +ε}，\ \ \] $ 0 $ 0 $ 0 $。因此，我们的主要结果在上述估计值上有所改善，同时也将其推广到$ s $的较大值和$ψ（x）$的更广泛的选择。 我们估计的新颖性是它们仅依赖于$ d $，$ s $和$ | a | $，并且独立于$ a $的直径。因此，当$ a $是一个稀疏集时，我们的结果比通过解耦和有效的一致性等方法提供的相应界限要强。因此，我们的策略与这两条方法不同，我们采用了发病率几何形状，算术组合学和分析数理论的技术。在其他应用中，我们的估计值导致稀疏序列的更强离散限制估计。
Let $s \geq 3$ be a natural number, let $ψ(x)$ be a polynomial with real coefficients and degree $d \geq 2$, and let $A$ be some large, non-empty, finite subset of real numbers. We use $E_{s,2}(A)$ to denote the number of solutions to the system of equations \[ \sum_{i=1}^{s} (ψ(x_i) - ψ(x_{i+s}) )= \sum_{i=1}^{s} ( x_i - x_{i+s} ) = 0, \] where $x_i \in A$ for each $1 \leq i \leq 2s$. Our main result shows that \[ E_{s,2}(A) \ll_{d,s} |A|^{2s -3 + η_{s}}, \] where $η_3 = 1/2$, and $η_{s} = (1/4- 1/7246)\cdot 2^{-s + 4}$ when $s \geq 4$. The only other previously known result of this flavour is due to Bourgain and Demeter, who showed that when $ψ(x) = x^2$ and $s=3$, we have \[E_{3,2}(A) \ll_ε |A|^{3 + 1/2 + ε},\] for each $ε> 0$. Thus our main result improves upon the above estimate, while also generalising it for larger values of $s$ and more wide-ranging choices of $ψ(x)$. The novelty of our estimates is that they only depend on $d$, $s$ and $|A|$, and are independent of the diameter of $A$. Thus when $A$ is a sparse set, our results are stronger than the corresponding bounds that are provided by methods such as decoupling and efficient congruencing. Consequently, our strategy differs from these two lines of approach, and we employ techniques from incidence geometry, arithmetic combinatorics and analytic number theory. Amongst other applications, our estimates lead to stronger discrete restriction estimates for sparse sequences.
