James Wright: Harmonic Analysis & Number Theory: Connections between Fourier Restriction and the Vinogradov Mean Value Theorem

Abstract

In this talk we explore the relationship between the Fourier Restriction problem and the Vinogradov Mean Value Theorem. These connections persist outside the euclidean setting and hold in any local field. In a certain range of exponents, the connections go in both directions, a Fourier restriction bound gives a precise count of approximate solutions to a system of Diophantine equations. On the other hand, showing the only approximate solutions are approximately trivial implies the Fourier restriction bound. This is joint work with Jonathan Hickman.