Higher dimensional fractal uncertainty

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.

Name thesis:degree_name

Doctoral

Department dc:contributor.department

Massachusetts Institute of Technology. Department of Mathematics

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2025

Author dc:creator

• Cohen, Alex

Advisor dc:contributor.advisor

• Guth, Larry

Statement dc:rights

• In Copyright - Educational Use Permitted
• Copyright retained by author(s)

Licence dc:rights.uri

• https://rightsstatements.org/page/InC-EDU/1.0/

Handle dc:identifier.uri

https://hdl.handle.net/1721.1/159893

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/159893