论文标题
不均匀vinogradov系统中的亚凸度
Subconvexity in inhomogeneous Vinogradov systems
论文作者
论文摘要
当$ k $和$ s $是自然数字，而$ \ mathbf h \ in \ mathbb z^k $，用$ j_ {s，k}（x; \ mathbf h）$表示系统的积分解决方案\ [\ sum_ = sum_ = 1}} S（x_i^j-y___i^j h__j j h_j j） $ 1 \ le x_i，y_i \ le x $。当$ s <k（k+1）/2 $和$（h_1，\ ldots，h_ {k-1}）\ ne {\ mathbf 0} $时，布兰德斯和休斯表明$ j_ {s，k}（x; \ mathbf h）= o（x^s）$。在本文中，我们改进了该结果的定量方面，并且在Vinogradov的平均值定理中的主要猜想扩展下，我们获得了$ j_ {s，k}（x; \ mathbf h）$的渐近公式，在关键情况下$ s = k+1）/2 $。后者需要次要的弧估计，而不是平方根取消。
When $k$ and $s$ are natural numbers and $\mathbf h\in \mathbb Z^k$, denote by $J_{s,k}(X;\mathbf h)$ the number of integral solutions of the system \[ \sum_{i=1}^s(x_i^j-y_i^j)=h_j\quad (1\le j\le k), \] with $1\le x_i,y_i\le X$. When $s<k(k+1)/2$ and $(h_1,\ldots ,h_{k-1})\ne {\mathbf 0}$, Brandes and Hughes have shown that $J_{s,k}(X;\mathbf h)=o(X^s)$. In this paper we improve on quantitative aspects of this result, and, subject to an extension of the main conjecture in Vinogradov's mean value theorem, we obtain an asymptotic formula for $J_{s,k}(X;\mathbf h)$ in the critical case $s=k(k+1)/2$. The latter requires minor arc estimates going beyond square-root cancellation.
