This work investigates the spectrality of self-affine measure <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1763_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="40" /> </InlineMediaObject> <EquationSource Format="TEX">$\mu _{M,D}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>μ</mi> <mrow> <mi>M</mi> <mo>,</mo> <mi>D</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> generated by an expanding matrix <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1763_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="87" /> </InlineMediaObject> <EquationSource Format="TEX">$M\in M_{3}(\mathbb {Z})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo>∈</mo> <msub> <mi>M</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and a four-element digit sets <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1763_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">$D\subset \mathbb {Z}^{3}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mn>3</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>. We introduce the transformation of left equivalence about the pair (M, D), which helps us keep the integer properties of M and D simultaneously, and provides new insights into studying the spectrality of self-affine measures in <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2025_1763_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^{n}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. The application of this method extends the known results in a simple manner.

Spectrality of Self-affine Measures with Four-element Digit Sets in $\mathbb {R}^{3}$

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发表日期：

2025年7月16日

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Ming-Shu Yang,
Jian-Wei Er

摘要

This work investigates the spectrality of self-affine measure $\mu _{M,D}$ $μ_{M, D}$ generated by an expanding matrix $M\in M_{3}(\mathbb {Z})$ $M \in M_{3} (Z)$ and a four-element digit sets $D\subset \mathbb {Z}^{3}$ $D \subset Z^{3}$ . We introduce the transformation of left equivalence about the pair (M, D), which helps us keep the integer properties of M and D simultaneously, and provides new insights into studying the spectrality of self-affine measures in $\mathbb {R}^{n}$ $R^{n}$ . The application of this method extends the known results in a simple manner.

Spectrality of Self-affine Measures with Four-element Digit Sets in \(\mathbb {R}^{3}\)

摘要