论文标题
沿平滑曲线的两参数家族Weyl和的混合界限
Hybrid bounds on two-parametric family Weyl sums along smooth curves
论文作者
论文摘要
我们获得了$（τx+c）ω（n）+xn $，$ n = 1，2，\ ldots $的$ k \ ge 2 $多项式的Weyl和weyl总和的新限制，并带有固定的$ω（t）\ in \ in \ in \ in \ in \ mathbb {z} [z} [t] $和$τ\ in \ mathbb in \ mathbb} [0,1）$和[0,1）$的所有$ x \。我们改善并推广了M.〜B。〜Erdogan和G.〜Shakan（2019）的一些最新结果，他们的工作还显示了这个问题与某些经典偏微分方程之间的联系。我们将其扩展到全部$（x，y）\ in [0,1）^2 $的多项式家庭$ xn+yΩ（n）$的更通用设置，该$ f（x，y）= z $用于一组$ z \ in [0,1）$ in [0,1）$ in [0,1）$ in [0,1）$ full full lebesgue度量，但前提是$ f $ f $ f $ f $ is hom f $ is hor houlder is houlder is houlder function。
We obtain a new bound on Weyl sums with degree $k\ge 2$ polynomials of the form $(τx+c) ω(n)+xn$, $n=1, 2, \ldots$, with fixed $ω(T) \in \mathbb{Z}[T]$ and $τ\in \mathbb{R}$, which holds for almost all $c\in [0,1)$ and all $x\in [0,1)$. We improve and generalise some recent results of M.~B.~Erdogan and G.~Shakan (2019), whose work also shows links between this question and some classical partial differential equations. We extend this to more general settings of families of polynomials $xn+y ω(n)$ for all $(x,y)\in [0,1)^2$ with $f(x,y)=z$ for a set of $z \in [0,1)$ of full Lebesgue measure, provided that $f$ is some Hölder function.
