Higher dimensional fractal uncertainty

Abstract

We prove that if a fractal set in Rᵈ avoids lines in a certain quantitative sense, which we call line porosity, then it has a fractal uncertainty principle. The main ingredient is a new higher dimensional Beurling and Malliavin multiplier theorem, which allows us to construct band-limited functions that decay rapidly on line porous sets. To prove this theorem, we first explicitly construct certain plurisubharmonic functions on Cᵈ. Then, following Bourgain, we use Hörmander’s L² theory for the ¯∂ equation to construct band-limited functions. The main theorem has since been applied by Kim and Miller to lower bounds for the mass of eigenfunctions on higher dimensional hyperbolic manifolds.