Translation invariance, exponential sums, and Waring's problem

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.

Submitted to Proceedings of the ICM 2014

References in corpus (3)

• The cubic case of the main conjecture in Vinogradov's mean value theorem
• Mean value estimates for odd cubic Weyl sums
• Diophantine methods for exponential sums, and exponential sums for Diophantine problems

Cited by in corpus (5)

• The cubic case of the main conjecture in Vinogradov's mean value theorem
• Nested efficient congruencing and relatives of Vinogradov's mean value theorem
• The Cubic Case of Vinogradov's Mean Value Theorem --- A Simplified Approach to Wooley's "Efficient Congruencing"
• Counting monochromatic solutions to diagonal Diophantine equations
• Effective $l^2$ decoupling for the parabola