Perturbations of Weyl sums

TL;DR

This paper investigates the behavior of Weyl sums for polynomials of degree at least three, establishing bounds that hold for almost all parameter choices and improving previous results for higher degrees.

Contribution

It proves that for almost all coefficient vectors, Weyl sums are bounded by a specific power of X, improving existing bounds for degrees five and above.

Findings

Established bounds for Weyl sums for almost all parameters.

Improved previous bounds for degrees k ≥ 5.

Demonstrated full measure set of parameters with desired properties.

Abstract

Write $f_k({\boldsymbol \alpha};X)=\sum_{x\le X}e(\alpha_1x+\ldots +\alpha_kx^k)$ $(k\ge 3)$. We show that there is a set ${\mathfrak B}\subseteq [0,1)^{k-2}$ of full measure with the property that whenever $(\alpha_2,\ldots ,\alpha_{k-1})\in {\mathfrak B}$ and $X$ is sufficiently large, then $$\sup_{(\alpha_1,\alpha_k)\in [0,1)^2}|f_k({\boldsymbol \alpha};X)|\le X^{1/2+4/(2k-1)}.$$ For $k\ge 5$, this improves on work of Flaminio and Forni, in which a Diophantine condition is imposed on $\alpha_k$, and the exponent of $X$ is $1-2/(3k(k-1))$.