On $L^{12}$ square root cancellation for exponential sums associated with nondegenerate curves in ${\mathbb R}^4$

TL;DR

This paper establishes sharp $L^{12}$ bounds for exponential sums linked to nondegenerate real analytic curves in four-dimensional space, advancing understanding of the Lindel"of hypothesis through novel analytical techniques.

Contribution

It introduces a new framework combining decoupling and quadratic Weyl sum estimates to achieve sharp bounds for exponential sums on nondegenerate curves in ${ m R}^4$.

Findings

Proved sharp $L^{12}$ estimates for exponential sums.

Extended Bourgain's progress on the Lindel"of hypothesis.

Developed a unified approach using decoupling and Weyl sum techniques.

Abstract

We prove sharp $L^{12}$ estimates for exponential sums associated with nondegenerate curves in ${\mathbb R}^4$. We place Bourgain's progress on the Lindel"of hypothesis in a larger framework that contains a continuum of estimates of different flavor. We enlarge the spectrum of methods by combining decoupling with quadratic Weyl sum estimates, to address new cases of interest. All results are proved in the general framework of real analytic curves.