Hybrid Bounds on Two-Parametric Families of Weyl Sums Along Smooth Curves

Abstract

We obtain a new bound on Weyl sums with degree $\mathit{k}\ge 2$ polynomials of the form $(\mathit{\tau }\mathit{x}+\mathit{c})\mathit{\omega }(\mathit{n})+\mathit{x}\mathit{n}$, $\mathit{n}=1,2,\dots$, with fixed $\mathit{\omega }(\mathit{T})\in \mathbb{Z}[\mathit{T}]$ and $\mathit{\tau }\in \mathbb{R}$, which holds for almost all $\mathit{c}\in [0,1)$ and all $\mathit{x}\in [0,1)$. We improve and generalize some recent results of Erdoǧan and 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 $\mathit{x}\mathit{n}+\mathit{y}\mathit{\omega }(\mathit{n})$ for all ${(\mathit{x},\mathit{y})\in [0,1)}^2$ with $\mathit{f}(\mathit{x},\mathit{y})=\mathit{z}$ for a set of $\mathit{z}\in [0,1)$ of full Lebesgue measure, provided that f is a Hölder function.