有限哈达玛对的有理谱及实直线上的有界谱集
Rational Spectra for Finite Hadamard Pairs and Bounded Spectral Sets on the real line
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中文总结 AI 辅助
研究有限谱对\((A,\Gamma)\)等的有理谱问题，核心方法是利用广义范德蒙德系统的模刚性定理等，主要贡献是证明相关谱为有理谱，完善等价关系，简化一维富格莱德猜想。
中文摘要 AI 辅助
我们证明了每个有限谱对\((A,\Gamma)\)（其中\(A\subset\mathbb Z\)，\(\Gamma\subset\mathbb R/\mathbb Z\)且\(0\in\Gamma\)）都有一个有理谱，即\(\Gamma\subset\mathbb Q/\mathbb Z\)。我们的论证依赖于由双曲正定核构造的满足逆正交关系的广义范德蒙德系统的模刚性定理。结合伽罗瓦共轭和克罗内克定理，这种刚性迫使相关的指数节点为单位根。作为应用，利用一维谱的周期性和纤维化，我们表明每个有界可测谱集\(\Omega\subset\mathbb R\)（\(|\Omega| = 1\)且\(0\in\Lambda\)）的谱\(\Lambda\)都包含在\(\mathbb Q\)中。这个有理谱结果完善了Dutkay和Lai证明的一系列等价关系，将完整的一维富格莱德猜想简化为其在\(\mathbb{Z}_n\)上的有限循环类似物。
英文摘要
We prove that every finite spectral pair \((A,Γ)\) with \(A\subset\mathbb Z\), \(Γ\subset\mathbb R/\mathbb Z\), and \(0\inΓ\) has a rational spectrum, that is, \(Γ\subset\mathbb Q/\mathbb Z\). Our argument relies on a modulus rigidity theorem for generalizedVandermonde systems satisfying inverse-orthogonality relations, constructed by hyperbolic positive-definite kernels. Combined with Galois conjugation and Kronecker's theorem, this rigidity forces the associated exponential nodes to be roots of unity. As an application, using the periodicity and fiberization of one-dimensional spectra, we show that every spectrum \(Λ\) of a bounded measurable spectral set \(Ω\subset\mathbb R\) with \(|Ω|=1\) and \(0\inΛ\) is contained in \(\mathbb Q\). This rational-spectrum result completes a chain of equivalences proven by Dutkay and Lai, which reduces the full one-dimensional Fuglede's conjecture to its finite cyclic analogues over \(\mathbb{Z}_n\).
