A hitchhiker’s guide to the Kakeya conjecture

Abstract

What is the dimension of a set in that contains a unit line segment in every direction? This is the question of the Kakeya conjecture, an open problem in geometric measure theory for n>3 with deep and surprising connections to Fourier and harmonic analysis. This thesis presents an elementary exposition of the Kakeya conjecture and related problems in analysis, intended for readers without advanced mathematical background. We discuss the construction of Kakeya sets, the proof of the Kakeya conjecture in the n=2 and finite field cases, Fefferman’s ball multiplier problem, and the restriction conjecture.