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arXiv · 2206.14131

Fractal uncertainty for discrete 2D Cantor sets

Alex Cohen
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Abstract

We prove that a self-similar Cantor set in $\mathbb{Z}_N \times \mathbb{Z}_N$ 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 Beukers & Smyth. Our theorem answers a question of Dyatlov and has applications to open quantum maps.

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Alex Cohen. 2022-06-28. Fractal uncertainty for discrete 2D Cantor sets. https://doi.org/10.2140/apde.2025.18.743

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