Fractal uncertainty for discrete 2D Cantor sets
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866913718039740416 |
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| author | Cohen, Alex |
| author_facet | Cohen, Alex |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_14131 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Fractal uncertainty for discrete 2D Cantor sets Cohen, Alex Classical Analysis and ODEs Analysis of PDEs Spectral Theory 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. |
| title | Fractal uncertainty for discrete 2D Cantor sets |
| topic | Classical Analysis and ODEs Analysis of PDEs Spectral Theory |
| url | https://arxiv.org/abs/2206.14131 |