{"title":"自仿射谱测度的任意稀疏谱","authors":"L.-X. An, C.-K. Lai","doi":"10.1007/s10476-023-0191-9","DOIUrl":null,"url":null,"abstract":"<div><p>Given an expansive matrix <i>R</i> ∈ <i>M</i><sub><i>d</i></sub>(ℤ) and a finite set of digit <i>B</i> taken from ℤ<sup><i>d</i></sup>/<i>R</i>(<i>ℤ</i><sup><i>d</i></sup>). It was shown previously that if we can find an <i>L</i> such that (<i>R, B, L</i>) forms a Hadamard triple, then the associated fractal self-affine measure generated by (<i>R, B</i>) admits an exponential orthonormal basis of certain frequency set Λ, and hence it is termed as a spectral measure. In this paper, we show that if #<i>B</i> < ∣det(<i>R</i>)∣, not only it is spectral, we can also construct arbitrarily sparse spectrum Λ in the sense that its Beurling dimension is zero.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"7","resultStr":"{\"title\":\"Arbitrarily Sparse Spectra for Self-Affine Spectral Measures\",\"authors\":\"L.-X. An, C.-K. Lai\",\"doi\":\"10.1007/s10476-023-0191-9\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Given an expansive matrix <i>R</i> ∈ <i>M</i><sub><i>d</i></sub>(ℤ) and a finite set of digit <i>B</i> taken from ℤ<sup><i>d</i></sup>/<i>R</i>(<i>ℤ</i><sup><i>d</i></sup>). It was shown previously that if we can find an <i>L</i> such that (<i>R, B, L</i>) forms a Hadamard triple, then the associated fractal self-affine measure generated by (<i>R, B</i>) admits an exponential orthonormal basis of certain frequency set Λ, and hence it is termed as a spectral measure. In this paper, we show that if #<i>B</i> < ∣det(<i>R</i>)∣, not only it is spectral, we can also construct arbitrarily sparse spectrum Λ in the sense that its Beurling dimension is zero.</p></div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2023-02-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"7\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10476-023-0191-9\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10476-023-0191-9","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Arbitrarily Sparse Spectra for Self-Affine Spectral Measures
Given an expansive matrix R ∈ Md(ℤ) and a finite set of digit B taken from ℤd/R(ℤd). It was shown previously that if we can find an L such that (R, B, L) forms a Hadamard triple, then the associated fractal self-affine measure generated by (R, B) admits an exponential orthonormal basis of certain frequency set Λ, and hence it is termed as a spectral measure. In this paper, we show that if #B < ∣det(R)∣, not only it is spectral, we can also construct arbitrarily sparse spectrum Λ in the sense that its Beurling dimension is zero.
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