On the spectrality of the non-homogeneous golden-mean self-similar measure
Abstract
We investigate the spectral properties of a class of inhomogeneous self-similar measures, which does not admit a non-trivial infinite convolution structure. A central example is the golden-mean self-similar measure μ, for which the existence of an exponential orthonormal basis in the associated L2-space has remained a long-standing open problem. The usual approach for homogeneous self-similar measures does not apply here, new methods are required. We establish several basic properties of the measure and then carry out a detailed numerical study of the zero set of its Fourier transform. Using a scanning and refinement algorithm that combines uniform grid sampling, quadratic interpolation, and golden-section search, we examine a wide range and find no real zeros of μ̂, which provides concrete evidence that μ is very likely non-spectral, suggesting that inhomogeneity may serve as a natural obstruction to the existence of exponential orthonormal bases. To the best of our knowledge, our paper is the first attempt to study the spectrality of such measures through a combined analytic and numerical framework.
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BibTeX
@misc{mao2026spectrality,
title = {{On the spectrality of the non-homogeneous golden-mean self-similar measure}},
author = {Yi-Qiu Mao and Zhi-Yi Wu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04701},
eprint = {2609.04701},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04701},
}
RIS
TY - UNPB
TI - On the spectrality of the non-homogeneous golden-mean self-similar measure
AU - Yi-Qiu Mao
AU - Zhi-Yi Wu
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.04701
UR - https://arxiv.org/abs/2609.04701
AB - We investigate the spectral properties of a class of inhomogeneous self-similar measures, which does not admit a non-trivial infinite convolution structure. A central example is the golden-mean self-similar measure μ, for which the existence of an exponential orthonormal basis in the associated L^2-space has remained a long-standing open problem. The usual approach for homogeneous self-similar measures does not apply here, new methods are required. We establish several basic properties of the measure and then carry out a detailed numerical study of the zero set of its Fourier transform. Using a scanning and refinement algorithm that combines uniform grid sampling, quadratic interpolation, and golden-section search, we examine a wide range and find no real zeros of μ̂, which provides concrete evidence that μ is very likely non-spectral, suggesting that inhomogeneity may serve as a natural obstruction to the existence of exponential orthonormal bases. To the best of our knowledge, our paper is the first attempt to study the spectrality of such measures through a combined analytic and numerical framework.
ER -