Hybrid bounds on two-parametric family Weyl sums along smooth curves

TL;DR

This paper establishes new bounds on Weyl sums involving two-parameter polynomial families along smooth curves, extending previous results and connecting to PDEs, with implications for almost all parameters in specified ranges.

Contribution

It introduces improved bounds on Weyl sums for two-parameter polynomial families along smooth curves, generalizing prior work and linking to partial differential equations.

Findings

New bounds on Weyl sums for almost all parameters

Extension to more general polynomial families

Connections to classical PDEs

Abstract

We obtain a new bound on Weyl sums with degree $k\ge 2$ polynomials of the form $(\tau x+c) \omega(n)+xn$, $n=1, 2, \ldots$, with fixed $\omega(T) \in \mathbb{Z}[T]$ and $\tau \in \mathbb{R}$, 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 $xn+y \omega(n)$ 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 H\"older function.