Higher dimensional fractal uncertainty
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cohen-alexcoh-phd-math-2025-thesis.pdf
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Thesis PDF
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Author(s)
Cohen, Alex
Advisor(s)
Guth, Larry
Date Issued
May 2025
Publisher
Massachusetts Institute of Technology
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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