Abstract
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We prove that a self-similar Cantor set in
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.
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Keywords
fractal uncertainty principle, quantum chaos, Lang
conjecture
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Mathematical Subject Classification
Primary: 28A80, 43A32, 81Q12
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Milestones
Received: 21 October 2022
Revised: 28 September 2023
Accepted: 21 November 2023
Published: 3 March 2025
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| © 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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