Decoupling inequalities and some mean-value theorems

TL;DR

This paper applies decoupling theory to diophantine problems, introducing a new mean value theorem that enhances estimates related to exponential sums in number theory.

Contribution

It presents a novel mean value theorem that advances the application of decoupling inequalities to diophantine and exponential sum problems.

Findings

New mean value theorem established

Improved estimates for exponential sums

Enhanced understanding of decoupling in number theory

Abstract

The purpose of this paper is to present some further applications of the general decoupling theory from [B-D1, 2] to certain diophantine issues. In particular, we concider mean value estimates relevant to the Bombieri-Iwaniec approach to exponential sums and arising in the work of Robert and Sargos [R-S]. Our main input is a new mean value theorem.