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Fractal uncertainty for discrete two-dimensional Cantor sets

Alex Cohen
Vol. 18 (2025), No. 3, 743–772
DOI: 10.2140/apde.2025.18.743
Abstract

We prove that a self-similar Cantor set in ℤN × ℤN has a fractal uncertainty principle if and only if it does not contain a pair of orthogonal lines. The key ingredient in our proof is a quantitative form of Lang’s conjecture in number theory due to Ruppert and to Beukers and Smyth. Our theorem answers a question of Dyatlov and has applications to open quantum maps.

Keywords
fractal uncertainty principle, quantum chaos, Lang conjecture
Mathematical Subject Classification
Primary: 28A80, 43A32, 81Q12
Milestones
Received: 21 October 2022
Revised: 28 September 2023
Accepted: 21 November 2023
Published: 3 March 2025
Authors
Alex Cohen
Massachusetts Institute of Technology
Cambridge, MA
United States
© 2025 The Author(s), under exclusive license to MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY).

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