发表机构
Central China Normal University; China University of Mining and Technology (Beijing)（华中师范大学; 中国矿业大学（北京））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文给出自复制平移集成为自相似测度谱的充要条件，推广了先前在实数域上的结果，并证明对偶谱集猜想对此类平移集成立。
AI 中文摘要
若指数函数族$\{e^{2\pi i \langle\lambda, x\rangle}:\lambda\in\Lambda\}$构成$L^2(\mu)$的一组标准正交基，则称离散集$\Lambda$为Borel概率测度$\mu$的一个谱。本文首先给出自复制平移集成为与乘积形式Hadamard三元组相关的自相似测度之谱的充分必要条件，该三元组由第一作者与Lai [Adv. Math. 431 (2023) Paper No.109257]发现。这些结果推广了Łaba和Wang [J. Funct. Anal. 193 (2002) 409-420]以及Dutkay和Lai [J. Math. Pures Appl. 107 (2017) 183-204]在$\mathbb R$中的研究。作为应用，我们能够显式构造这样的自复制谱。最后，我们证明对偶谱集猜想对自复制平移集成立。
英文摘要
A discrete set $Λ$ is called a {\it spectrum} of a Borel probability measure $μ$ if the exponential functions $\{e^{2πi \langleλ, x\rangle}:λ\inΛ\}$ form an orthonormal basis for $L^2(μ)$. In this work, we first give necessary and sufficient conditions for a self-replicating translation set to be a spectrum of the self-similar measure associated with a product-form Hadamard triple, which were discovered by the first-named author and Lai [Adv. Math. 431 (2023) Paper No.109257]. These results extend the studies by Łaba and Wang [J. Funct. Anal. 193 (2002) 409-420], Dutkay and Lai [J. Math. Pures Appl. 107 (2017) 183-204] in $\mathbb R$. As an application, we can explicitly construct such a self-replicating spectrum. Finally, we prove that the dual spectral set conjecture holds for a self-replicating translation set.
