For integers <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$q_k\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>q</mi> <mi>k</mi> </msub> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\{D_k=\{0,1,\cdots ,q_k-1\}\}_{k=1}^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mo stretchy="false">{</mo> <msub> <mi>D</mi> <mi>k</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>q</mi> <mi>k</mi> </msub> <mo>-</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> </math></EquationSource> </InlineEquation> be a sequence of digit sets and let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\{b_k=q_kr_k\}_{k=1}^\infty $</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mrow> <mo stretchy="false">{</mo> <msub> <mi>b</mi> <mi>k</mi> </msub> <mo>=</mo> <msub> <mi>q</mi> <mi>k</mi> </msub> <msub> <mi>r</mi> <mi>k</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>∞</mi> </msubsup> </math></EquationSource> </InlineEquation> be a sequences of integers. An et al. [<CitationRef CitationID="CR4">4</CitationRef>] have shown that the infinite convolution <Equation ID="Equ26"> <EquationSource Format="TEX">$\begin{aligned} \mu _{\{b_k\},\{D_k\}}=\delta _{b_1^{-1}D_1}\delta _{(b_1b_2)^{-1}D_2}\delta _{(b_1b_2b_3)^{-1}D_3} \cdots \delta _{(b_1b_2\cdots b_k)^{-1}D_k}*\cdots \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msub> <mi>μ</mi> <mrow> <mrow> <mo stretchy="false">{</mo> <msub> <mi>b</mi> <mi>k</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mo>,</mo> <mrow> <mo stretchy="false">{</mo> <msub> <mi>D</mi> <mi>k</mi> </msub> <mo stretchy="false">}</mo> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>δ</mi> <mrow> <msubsup> <mi>b</mi> <mn>1</mn> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>D</mi> <mn>1</mn> </msub> </mrow> </msub> <mrow /> <mo>∗</mo> <msub> <mi>δ</mi> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mn>1</mn> </msub> <msub> <mi>b</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>D</mi> <mn>2</mn> </msub> </mrow> </msub> <mrow /> <mo>∗</mo> <msub> <mi>δ</mi> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mn>1</mn> </msub> <msub> <mi>b</mi> <mn>2</mn> </msub> <msub> <mi>b</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>D</mi> <mn>3</mn> </msub> </mrow> </msub> <mrow /> <mo>∗</mo> <mo>⋯</mo> <mrow /> <mo>∗</mo> <msub> <mi>δ</mi> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>b</mi> <mn>1</mn> </msub> <msub> <mi>b</mi> <mn>2</mn> </msub> <mo>⋯</mo> <msub> <mi>b</mi> <mi>k</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msub> <mi>D</mi> <mi>k</mi> </msub> </mrow> </msub> <mrow /> <mo>∗</mo> <mo>⋯</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>is a spectral measure. In this paper, we consider the spectral eigenvalue problem for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mu _{\{b_k\},\{D_k\}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>μ</mi> <mrow> <mrow> <mo stretchy="false">{</mo> <msub> <mi>b</mi> <mi>k</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mo>,</mo> <mrow> <mo stretchy="false">{</mo> <msub> <mi>D</mi> <mi>k</mi> </msub> <mo stretchy="false">}</mo> </mrow> </mrow> </msub> </math></EquationSource> </InlineEquation>, i.e., we seek conditions for real number t such that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$t\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mi mathvariant="normal">Λ</mi> </mrow> </math></EquationSource> </InlineEquation> is also a spectrum of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mu _{\{b_k\},\{D_k\}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>μ</mi> <mrow> <mrow> <mo stretchy="false">{</mo> <msub> <mi>b</mi> <mi>k</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mo>,</mo> <mrow> <mo stretchy="false">{</mo> <msub> <mi>D</mi> <mi>k</mi> </msub> <mo stretchy="false">}</mo> </mrow> </mrow> </msub> </math></EquationSource> </InlineEquation> for some spectrum <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation>. We deonstrate that t is a spectral eigenvalue of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mu _{\{b_k\},\{D_k\}}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>μ</mi> <mrow> <mrow> <mo stretchy="false">{</mo> <msub> <mi>b</mi> <mi>k</mi> </msub> <mo stretchy="false">}</mo> </mrow> <mo>,</mo> <mrow> <mo stretchy="false">{</mo> <msub> <mi>D</mi> <mi>k</mi> </msub> <mo stretchy="false">}</mo> </mrow> </mrow> </msub> </math></EquationSource> </InlineEquation> only if <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$t=\frac{p}{q}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>=</mo> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation> for some integers p, q with <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\gcd (p,q)=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">gcd</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\gcd (p,q_k)=\gcd (q,q_k)=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">gcd</mo> <mrow> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <msub> <mi>q</mi> <mi>k</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo movablelimits="true">gcd</mo> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <msub> <mi>q</mi> <mi>k</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$k\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Moreover, if <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\{\frac{b_k}{q_k}\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mfrac> <msub> <mi>b</mi> <mi>k</mi> </msub> <msub> <mi>q</mi> <mi>k</mi> </msub> </mfrac> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> is unbounded, the necessary condition is also sufficient.

The Spectral Eigenvalues of a Class of Moran Measures with Continuous Digits

期刊论文
发表日期：

2026年2月6日

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Si Chen,
Jing-cheng Liu,
Ming Liu

摘要

For integers $q_k\ge 2$ $q_{k} \geq 2$ , let $\{D_k=\{0,1,\cdots ,q_k-1\}\}_{k=1}^\infty $ ${D_{k} = {0, 1, \dots, q_{k} - 1}}_{k = 1}^{\infty}$ be a sequence of digit sets and let $\{b_k=q_kr_k\}_{k=1}^\infty $ ${b_{k} = q_{k} r_{k}}_{k = 1}^{\infty}$ be a sequences of integers. An et al. [4] have shown that the infinite convolution $\begin{aligned} \mu _{\{b_k\},\{D_k\}}=\delta _{b_1^{-1}D_1}*\delta _{(b_1b_2)^{-1}D_2}*\delta _{(b_1b_2b_3)^{-1}D_3} *\cdots *\delta _{(b_1b_2\cdots b_k)^{-1}D_k}*\cdots \end{aligned}$ $\begin{matrix} μ_{{b_{k}}, {D_{k}}} = δ_{b_{1}^{- 1} D_{1}} * δ_{{(b_{1} b_{2})}^{- 1} D_{2}} * δ_{{(b_{1} b_{2} b_{3})}^{- 1} D_{3}} * \dots * δ_{{(b_{1} b_{2} \dots b_{k})}^{- 1} D_{k}} * \dots \end{matrix}$ is a spectral measure. In this paper, we consider the spectral eigenvalue problem for $\mu _{\{b_k\},\{D_k\}}$ $μ_{{b_{k}}, {D_{k}}}$ , i.e., we seek conditions for real number t such that $t\Lambda $ $t Λ$ is also a spectrum of $\mu _{\{b_k\},\{D_k\}}$ $μ_{{b_{k}}, {D_{k}}}$ for some spectrum $\Lambda $ $Λ$ . We deonstrate that t is a spectral eigenvalue of $\mu _{\{b_k\},\{D_k\}}$ $μ_{{b_{k}}, {D_{k}}}$ only if $t=\frac{p}{q}$ $t = \frac{p}{q}$ for some integers p, q with $\gcd (p,q)=1$ $gcd (p, q) = 1$ and $\gcd (p,q_k)=\gcd (q,q_k)=1$ $gcd (p, q_{k}) = gcd (q, q_{k}) = 1$ for all $k\ge 1$ $k \geq 1$ . Moreover, if $\{\frac{b_k}{q_k}\}$ ${\frac{b_{k}}{q_{k}}}$ is unbounded, the necessary condition is also sufficient.