论文标题
特征功能的Beurling-Wintner问题
The Beurling-Wintner problem for characteristic functions
论文作者
论文摘要
本文涉及Beurling和Wintner提出的一个长期存在的问题，即扩张系统的完整性$ \ {φ（kx）：k = 1,2，\ cdots \} $由$ \ mathbb {r} $ in l^2 [0,1] $ in $ \ mathbb {r} $ on $ \ mathbb {r} $产生的。到目前为止，即使对于特征功能，也没有明确描述Beurling-Wintner问题的解决方案。我们专注于特征函数$ \ mathbf {1} _v $的开放子集$ v $的$（0,1）$，其中$ v $是有限的许多间隔的结合，具有合理的端点。使用分析数理论的基本技术，我们在最有趣的情况下充分解决了Beurling-Wintner问题，并展示了这种$ V $的明确形式。结果，它为科兹洛夫问题的理性版本提供了完整的解决方案。此外，我们发现贝尔林 - 沃特纳的问题与双重猜想和Sophie Germain Prime猜想密切相关。
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.
