For a prime number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, let A be a diagonal expanding matrix, and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$0\in {\mathcal {B}}\subset {\mathbb {Z}}^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>∈</mo> <mi mathvariant="script">B</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> be a p-element digit set satisfying <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {Z}(\hat{\delta }_{{\mathcal {B}}})=\bigcup _{j=1}^{p-1} \left( \frac{j}{p}(\omega ,\rho )^t+{\mathbb {Z}}^2\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">Z</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mover accent="true"> <mi>δ</mi> <mo stretchy="false">^</mo> </mover> <mi mathvariant="script">B</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msubsup> <mo>⋃</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>p</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mfenced close=")" open="("> <mfrac> <mi>j</mi> <mi>p</mi> </mfrac> <msup> <mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>,</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> <mi>t</mi> </msup> <mo>+</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>2</mn> </msup> </mfenced> </mrow> </math></EquationSource> </InlineEquation> for some <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\{\rho ,\omega \}\subset \{1,\cdots ,p-1\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mi>ρ</mi> <mo>,</mo> <mi>ω</mi> <mo stretchy="false">}</mo> <mo>⊂</mo> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>p</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\gcd (\rho ,\omega )=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">gcd</mo> <mo stretchy="false">(</mo> <mi>ρ</mi> <mo>,</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. The associated self-affine measure <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mu _{A,{\mathcal {B}}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>μ</mi> <mrow> <mi>A</mi> <mo>,</mo> <mi mathvariant="script">B</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> is a spectral measure. In this paper, we investigate the spectral structure and spectral eigenmatrix problem associated with <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mu _{A,{\mathcal {B}}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>μ</mi> <mrow> <mi>A</mi> <mo>,</mo> <mi mathvariant="script">B</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>. Specifically, some sufficient and necessary conditions for a real diagonal matrix <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Re $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℜ</mi> </math></EquationSource> </InlineEquation> to be a spectral eigenmatrix of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mu _{A,{\mathcal {B}}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>μ</mi> <mrow> <mi>A</mi> <mo>,</mo> <mi mathvariant="script">B</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> are given under certain constraints on A and <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\Re $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℜ</mi> </math></EquationSource> </InlineEquation>. This extends some known results on the spectral eigenmatrix problem of planar self-affine measures.

Spectral eigenmatrix problem of planar self-affine measures with p digits

期刊论文
发表日期：

2026年7月9日

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Ming-Liang Chen,
Zhi-Long Chen,
Jian Cao

摘要

For a prime number $p\ge 3$ $p \geq 3$ , let A be a diagonal expanding matrix, and let $0\in {\mathcal {B}}\subset {\mathbb {Z}}^2$ $0 \in B \subset Z^{2}$ be a p-element digit set satisfying $\mathcal {Z}(\hat{\delta }_{{\mathcal {B}}})=\bigcup _{j=1}^{p-1} \left( \frac{j}{p}(\omega ,\rho )^t+{\mathbb {Z}}^2\right) $ $Z ({\hat{δ}}_{B}) = ⋃_{j = 1}^{p - 1} (\frac{j}{p} {(ω, ρ)}^{t} + Z^{2})$ for some $\{\rho ,\omega \}\subset \{1,\cdots ,p-1\}$ ${ρ, ω} \subset {1, \dots, p - 1}$ , where $\gcd (\rho ,\omega )=1$ $gcd (ρ, ω) = 1$ . The associated self-affine measure $\mu _{A,{\mathcal {B}}}$ $μ_{A, B}$ is a spectral measure. In this paper, we investigate the spectral structure and spectral eigenmatrix problem associated with $\mu _{A,{\mathcal {B}}}$ $μ_{A, B}$ . Specifically, some sufficient and necessary conditions for a real diagonal matrix $\Re $ $ℜ$ to be a spectral eigenmatrix of $\mu _{A,{\mathcal {B}}}$ $μ_{A, B}$ are given under certain constraints on A and $\Re $ $ℜ$ . This extends some known results on the spectral eigenmatrix problem of planar self-affine measures.