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Scaling spectrum of a class of self-similar measures with product form on ℝ

  • Shan-Feng Yi and Min-Min Zhang ORCID logo EMAIL logo
Published/Copyright: April 24, 2024
Forum Mathematicum
From the journal Forum Mathematicum Volume 37 Issue 3

Abstract

Let p, q, N ≥ 2 be three positive integers and let D = { 0 , 1 , … , N - 1 } ⊕ N p ⁢ { 0 , 1 , … , N - 1 } be a product form digit set. It is well known that if q ∤ p , then the self-similar measure μ N q , D generated by the iterated function system { ( N q ) - 1 ⁢ ( x + d ) } d ∈ D , x ∈ ℝ is a spectral measure with a spectrum

Λ ⁢ ( N q , C ) = { ∑ i = 0 finite c i ⁢ N q ⁢ i : c i ∈ C } ,

where C = N q - p - 1 ⁢ D . In this paper, based on the properties of cyclic groups in number theory, we give some conditions on real number t under which the scaling set t ⁢ Λ ⁢ ( N q , C ) is also a spectrum of μ N q , D .

Keywords: Self-similar measure; spectral measure; scaling spectra; product form
MSC 2020: 42C05; 42A65; 28A80

Communicated by Christopher D. Sogge

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12371087

Award Identifier / Grant number: 11971194

Funding statement: The work is supported by the National Natural Science Foundation of China (Grants No. 12371087 and No. 11971194).

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Received: 2023-12-18
Published Online: 2024-04-24
Published in Print: 2025-02-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Square-integrable representations and the coadjoint action of solvable Lie groups
  3. Deviation probabilities for extremal eigenvalues of large Chiral non-Hermitian random matrices
  4. Nef vector bundles on a hyperquadric with first Chern class two
  5. Scaling spectrum of a class of self-similar measures with product form on ℝ
  6. Idempotent decomposition of regularity and characterization for the accumulation of associated spectrum
  7. Geodesic orbit Randers metrics in homogeneous bundles over generalized Stiefel manifolds
  8. Proof of some conjectures of Guo and of Tang
  9. On Absolute and Quantitative Subspace Theorems
  10. Birational equivalence of the Zassenhaus varieties for basic classical Lie superalgebras and their purely-even reductive Lie subalgebras in odd characteristic
  11. Geometric approach to the Moore–Penrose inverse and the polar decomposition of perturbations by operator ideals
  12. The Weil bound for generalized Kloosterman sums of half-integral weight
  13. Relative Rota–Baxter groups and skew left braces
  14. Asai gamma factors over finite fields
  15. Representations of non-finitely graded Lie algebras related to Virasoro algebra
  16. Multiple solutions for fractional elliptic systems
  17. Arithmetic Bohr radius for the Minkowski space
  18. On Strichartz estimates for many-body Schrödinger equation in the periodic setting
https://doi.org/10.1515/forum-2023-0466
Keywords for this article
Self-similar measure; spectral measure; scaling spectra; product form

Articles in the same Issue

  1. Frontmatter
  2. Square-integrable representations and the coadjoint action of solvable Lie groups
  3. Deviation probabilities for extremal eigenvalues of large Chiral non-Hermitian random matrices
  4. Nef vector bundles on a hyperquadric with first Chern class two
  5. Scaling spectrum of a class of self-similar measures with product form on ℝ
  6. Idempotent decomposition of regularity and characterization for the accumulation of associated spectrum
  7. Geodesic orbit Randers metrics in homogeneous bundles over generalized Stiefel manifolds
  8. Proof of some conjectures of Guo and of Tang
  9. On Absolute and Quantitative Subspace Theorems
  10. Birational equivalence of the Zassenhaus varieties for basic classical Lie superalgebras and their purely-even reductive Lie subalgebras in odd characteristic
  11. Geometric approach to the Moore–Penrose inverse and the polar decomposition of perturbations by operator ideals
  12. The Weil bound for generalized Kloosterman sums of half-integral weight
  13. Relative Rota–Baxter groups and skew left braces
  14. Asai gamma factors over finite fields
  15. Representations of non-finitely graded Lie algebras related to Virasoro algebra
  16. Multiple solutions for fractional elliptic systems
  17. Arithmetic Bohr radius for the Minkowski space
  18. On Strichartz estimates for many-body Schrödinger equation in the periodic setting
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