The Beurling-Wintner problem for characteristic functions
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866917629972709376 |
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| author | Dan, Hui Guo, Kunyu |
| author_facet | Dan, Hui Guo, Kunyu |
| contents | This paper concerns a long-standing problem raised by Beurling and Wintner on completeness of the dilation system $\{φ(kx):k=1,2,\cdots\}$ generated by the odd periodic extension on $\mathbb{R}$ of any $φ\in L^2[0,1]$. Up to now there has been no explicit description of solutions of the Beurling-Wintner problem even for characteristic functions. We focus on characteristic function $\mathbf{1}_V$ of an open subset $V$ of $(0,1)$ where $V$ is the union of finitely many intervals with rational endpoints. Using substantially techniques from analytic number theory, we fully solved the Beurling-Wintner problem in most interesting situations and exhibit the explicit form of such $V$. As a consequence, it yields a complete solution for the rational version of Kozlov's problem. Moreover, we find that the Beurling-Wintner problem is closely related to the Twin Prime Conjecture and the Sophie Germain Prime Conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2005_09779 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The Beurling-Wintner problem for characteristic functions Dan, Hui Guo, Kunyu Classical Analysis and ODEs Complex Variables Number Theory This paper concerns a long-standing problem raised by Beurling and Wintner on completeness of the dilation system $\{φ(kx):k=1,2,\cdots\}$ generated by the odd periodic extension on $\mathbb{R}$ of any $φ\in L^2[0,1]$. Up to now there has been no explicit description of solutions of the Beurling-Wintner problem even for characteristic functions. We focus on characteristic function $\mathbf{1}_V$ of an open subset $V$ of $(0,1)$ where $V$ is the union of finitely many intervals with rational endpoints. Using substantially techniques from analytic number theory, we fully solved the Beurling-Wintner problem in most interesting situations and exhibit the explicit form of such $V$. As a consequence, it yields a complete solution for the rational version of Kozlov's problem. Moreover, we find that the Beurling-Wintner problem is closely related to the Twin Prime Conjecture and the Sophie Germain Prime Conjecture. |
| title | The Beurling-Wintner problem for characteristic functions |
| topic | Classical Analysis and ODEs Complex Variables Number Theory |
| url | https://arxiv.org/abs/2005.09779 |