Characterization on spectra of self-similar measures and the dual spectral set conjecture
Abstract
A discrete set Λ is called a {\it spectrum} of a Borel probability measure μ if the exponential functions {e2πi ⟨λ, x⟩:λ∈Λ} form an orthonormal basis for L2(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 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.
Related
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BibTeX
@misc{an2026characterization,
title = {{Characterization on spectra of self-similar measures and the dual spectral set conjecture}},
author = {Li-Xiang An and Yan-Song Fu and Ming-Xuan Jiang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05830},
eprint = {2609.05830},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05830},
}
RIS
TY - UNPB
TI - Characterization on spectra of self-similar measures and the dual spectral set conjecture
AU - Li-Xiang An
AU - Yan-Song Fu
AU - Ming-Xuan Jiang
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.05830
UR - https://arxiv.org/abs/2609.05830
AB - A discrete set Λ is called a {\it spectrum} of a Borel probability measure μ if the exponential functions {e^2πi ⟨λ, x⟩:λ∈Λ} 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 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.
ER -