Translation invariance, exponential sums, and Waring's problem

TL;DR

This paper advances mean value estimates for high-degree exponential sums using the efficient congruencing method, with applications to Weyl sums, polynomial distribution, and Waring's problem.

Contribution

It introduces a novel approach leveraging translation invariance to improve bounds on exponential sums, approaching conjectured optimal estimates.

Findings

Mean value estimates approach the best possible conjectured bounds

Application of efficient congruencing to polynomial distribution modulo 1

Progress on Waring's problem through improved exponential sum estimates

Abstract

We describe mean value estimates for exponential sums of degree exceeding 2 that approach those conjectured to be best possible. The vehicle for this recent progress is the efficient congruencing method, which iteratively exploits the translation invariance of associated systems of Diophantine equations to derive powerful congruence constraints on the underlying variables. There are applications to Weyl sums, the distribution of polynomials modulo 1, and other Diophantine problems such as Waring's problem.