Spectrality of generalized Sierpinski-type self-affine measures

Abstract

For an expanding integer matrix $M\in M_2(\mathbb{Z})$ and an integer digit set $D=\{(0,0{)}^t,({\alpha }_1,{\alpha }_2{)}^t,({\beta }_1,{\beta }_2{)}^t\}$ with ${\alpha }_1{\beta }_2-{\alpha }_2{\beta }_1\ne 0$, let ${\mu }_{M,D}$ be the Sierpinski-type self-affine measure defined by ${\mu }_{M,D}(\cdot )=\frac{1}{3}{\sum }_{d\in D}{\mu }_{M,D}(M(\cdot )-d)$. In [5.36], the authors separately investigated the spectral property of the measure ${\mu }_{M,D}$ in the case of $\det (M)\notin 3\mathbb{Z}$ or ${\alpha }_1{\beta }_2-{\alpha }_2{\beta }_1\notin 3\mathbb{Z}$. In this paper, we consider the remaining case where $\det (M)\in 3\mathbb{Z}$ and ${\alpha }_1{\beta }_2-{\alpha }_2{\beta }_1\in 3\mathbb{Z}$, and give the necessary and sufficient conditions for ${\mu }_{M,D}$ to be a spectral measure. This completely settles the spectrality of the Sierpinski-type self-affine measure ${\mu }_{M,D}$.

References in corpus (2)

• Complex Hadamard matrices and the Spectral Set Conjecture
• The spectrality of self-affine measure under the similarity transformation of $G{L}_n(p)$