论文标题
二进制形式的小部分部分
Small fractional parts of binary forms
论文作者
论文摘要
我们获得了形状$ψ（x，y）的二进制形式的分数部分的界限$α_k，α_l，\ ldots，α_0\ in \ Mathbb {r} $和$ l \ l \ leq k-2。$通过利用Vinogradov的最新进度，在Vinogradov的平均价值定理上的最新进度和早期在增量数字上的增值总和上的早期工作，我们在平稳数字上超过了这些$ $ $ $的$ k $ k $ k $ s $ s $ s $ s $ s $ s $ q \ begin {equination*} \ min _ {\ ordack {0 \ leq x，y \ leq x \\（x，y）
We obtain bounds on fractional parts of binary forms of the shape $$Ψ(x,y)=α_k x^k+α_l x^ly^{k-l}+α_{l-1}x^{l-1}y^{k-l+1}+\cdots+α_0 y^k$$ with $α_k,α_l,\ldots,α_0\in\mathbb{R}$ and $l\leq k-2.$ By exploiting recent progress on Vinogradov's mean value theorem and earlier work on exponential sums over smooth numbers, we derive estimates superior to those obtained hitherto for the best exponent $σ$, depending on $k$ and $l,$ such that \begin{equation*} \min_{\substack{0\leq x,y\leq X\\(x,y)\neq (0,0)}}\|Ψ(x,y)\|\leq X^{-σ+ε}.\end{equation*}
