Spectral structure and eigenmatrices of self-affine measures with p-digits in $ \bf{\mathbb{R}^{n}} $

Abstract

For a prime number p > 2, let 0 ∈ D ⊂ ℤⁿ be a p-element digit set satisfying $ \mathcal{Z}(\widehatδ_D) =\cup_{j=1}^{p-1}(\frac{j}{p}\bm{a}+\mathbb{Z}^{n}) $ for some a ∈ { (i₁, …, i_n)^t : i_k ∈ [1, p−1] ∩ ℤ, 1≤ k≤ n }, where 𝒵(δ̂_D) is the zero set of the Fourier transform of δ_D. Let Q be an integer expansive diagonal matrix in ℝⁿ, the self-affine measure μ_Q,D is defined by μ_Q,D(·) = 1/#D ∑_{d ∈ D} μ_Q,D(Q(·) − d). In this paper, we first provide sufficient condition for a maximal orthogonal family E_Λ={ e^{−2πi ⟨ λ, x ⟩} : λ∈ Λ⊂ℝⁿ} to be an orthogonal basis of L²(μ_Q,D) when Q = pdiag[q, …, q] with q≥ 1. Then we obtain necessary and sufficient conditions for the integer matrix R such that E_Λ and E_RΛ are both orthogonal basis of L²(μ_Q,D). Furthermore, for Q = pdiag[l₁, …, l_n] with |l₁ … l_n| > 1, we give a necessary and sufficient condition under which the real diagonal matrix is the second type spectral eigenmatrices of μ_Q,D.

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BibTeX RIS

BibTeX

@misc{chen2026spectral,
  title = {{Spectral structure and eigenmatrices of self-affine measures with $p$-digits in $ \bf\mathbbR^n $}},
  author = {Jia-Long Chen and Wen-Hui Ai},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07468},
  eprint = {2609.07468},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07468},
}

RIS

TY  - UNPB
TI  - Spectral structure and eigenmatrices of self-affine measures with p-digits in $ \bf{\mathbb{R}^{n}} $
AU  - Jia-Long Chen
AU  - Wen-Hui Ai
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07468
UR  - https://arxiv.org/abs/2609.07468
AB  - For a prime number p > 2, let 0 ∈ D ⊂ ℤ^n be a p-element digit set satisfying $ \mathcal{Z}(\widehatδ_D) =\cup_{j=1}^{p-1}(\frac{j}{p}\bm{a}+\mathbb{Z}^{n}) $ for some a ∈ { (i_1, …, i_n)^t : i_k ∈ [1, p−1] ∩ ℤ, 1≤ k≤ n }, where 𝒵(δ̂_D) is the zero set of the Fourier transform of δ_D. Let Q be an integer expansive diagonal matrix in ℝ^n, the self-affine measure μ_Q,D is defined by μ_Q,D(·) = 1/#D ∑_d ∈ D μ_Q,D(Q(·) − d). In this paper, we first provide sufficient condition for a maximal orthogonal family E_Λ={ e^−2πi ⟨ λ, x ⟩ : λ∈ Λ⊂ℝ^n} to be an orthogonal basis of L^2(μ_Q,D) when Q = pdiag[q, …, q] with q≥ 1. Then we obtain necessary and sufficient conditions for the integer matrix R such that E_Λ and E_RΛ are both orthogonal basis of L^2(μ_Q,D). Furthermore, for Q = pdiag[l_1, …, l_n] with |l_1 … l_n| > 1, we give a necessary and sufficient condition under which the real diagonal matrix is the second type spectral eigenmatrices of μ_Q,D.
ER  -

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