发表机构
Tajik State University of Finance and Economics（塔吉克斯坦国立财经大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对$n\geq3$的短Weyl和在特定中间有理逼近范围得到一致点态估计，推导几乎成比例求和项的华林问题表示数的渐近公式，改进了$H$的容许下界。
AI 中文摘要
我们针对次数为$n\geq3$的短Weyl和，在中间有理逼近范围$\frac{1}{qx^{n-2}y}\ll|\lambda|\ll\frac{1}{qy^{n-1}}$内得到了一致点态估计。该范围源于推导几乎成比例求和项的华林问题渐近公式时，对剩余积分应用狄利克雷有理逼近定理的第二次应用。对于$r=2^n+1$及满足$\mu_1+\cdots+\mu_r=1$的固定正数$\mu_1,\ldots,\mu_r$，我们推导了满足$x_1^n+\cdots+x_r^n=N$、$|x_i^n-\mu_iN|\leq H$（$1\leq i\leq r$）的表示数的渐近公式，其有效范围为$N^{1-\theta(n,r)+\varepsilon}\leq H\leq\frac{N}{\ln N}$，其中$\theta(n,r)=\frac{2}{n\bigl((r-1)(n-1)+2\bigr)}$。所得$H$的容许下界对每个$n\geq3$都改进了之前的已知界。
英文摘要
We obtain a uniform pointwise estimate for short Weyl sums of degree $n\geq3$ in the intermediate rational-approximation range $$ \frac{1}{qx^{n-2}y}\ll|λ| \ll\frac{1}{qy^{n-1}}. $$ This range arises from a second application of Dirichlet's rational approximation theorem in estimating the residual integral that occurs in the derivation of an asymptotic formula for Waring's problem with almost proportional summands. For $r=2^n+1$ and fixed positive numbers $μ_1,\ldots,μ_r$ satisfying $$ μ_1+\cdots+μ_r=1, $$ we derive an asymptotic formula for the number of representations $$ x_1^n+\cdots+x_r^n=N, \qquad |x_i^n-μ_iN|\leq H,\qquad 1\le i \le r, $$ valid for $$ N^{1-θ(n,r)+\varepsilon}\le H \le \frac{N}{\ln N}, \quad θ(n,r)= \frac{2}{n\bigl((r-1)(n-1)+2\bigr)}. $$ The resulting admissible lower bound for $H$ improves the previously known bound for every $n\geq3$.
Comments17 pages, 1 table
