Speaker: Michael Christ (UC Berkeley)

Title: An extremal problem concerning Fourier coefficients

Abstract:
Consider a set in Euclidean space, and consider the $L^q$ norm of its Fourier transform. Among sets of specified measure, what is the largest value of this norm? Is it attained? If so, by which sets?

These natural questions seem to have received little attention. I will state several partial results, and indicate some of the ideas in the proofs. One ingredient is a compactness theorem, whose proof relies on an inverse theorem of additive combinatorics.