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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2406.06815 |
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| _version_ | 1866913385063383040 |
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| author | Kangabire, Alain |
| author_facet | Kangabire, Alain |
| contents | We prove two results on Fractal Uncertainty Principle (FUP) for discrete Cantor sets with large alphabets. First, we give an example of an alphabet with dimension $δ\in (\frac12,1)$ where the FUP exponent is exponentially small as the size of the alphabet grows. Secondly, for $δ\in (0,\frac12]$ we show that a similar alphabet has a large FUP exponent, arbitrarily close to the optimal upper bound of $\frac12-\frac\delta2$, if we dilate the Fourier transform by a factor satisfying a generic Diophantine condition. We give an application of the latter result to spectral gaps for open quantum baker's maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_06815 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds on the fractal uncertainty exponent and a spectral gap Kangabire, Alain Analysis of PDEs Spectral Theory We prove two results on Fractal Uncertainty Principle (FUP) for discrete Cantor sets with large alphabets. First, we give an example of an alphabet with dimension $δ\in (\frac12,1)$ where the FUP exponent is exponentially small as the size of the alphabet grows. Secondly, for $δ\in (0,\frac12]$ we show that a similar alphabet has a large FUP exponent, arbitrarily close to the optimal upper bound of $\frac12-\frac\delta2$, if we dilate the Fourier transform by a factor satisfying a generic Diophantine condition. We give an application of the latter result to spectral gaps for open quantum baker's maps. |
| title | Bounds on the fractal uncertainty exponent and a spectral gap |
| topic | Analysis of PDEs Spectral Theory |
| url | https://arxiv.org/abs/2406.06815 |