On spectral and non-spectral problem for the planar self-similar measures with four element digit sets

Abstract

Abstract We consider the self-similar measure μ M , 𝒟 {\mu_{M,{\mathcal{D}}}} generated by an expanding real matrix M = ( ρ - 1 0 0 ρ - 1 ) ∈ M 2 ⁢ ( ℝ ) {M=\begin{pmatrix}\rho^{-1}&0\\ 0&\rho^{-1}\end{pmatrix}\in M_{2}({\mathbb{R}})} and a digit set 𝒟 = { ( 0 0 ) , ( a b ) , ( c d ) , ( a + c b + d ) } ⊆ ℤ 2 . {{\mathcal{D}}=\Biggl{\{}\begin{pmatrix}0\\ 0\end{pmatrix},\begin{pmatrix}a\\ b\end{pmatrix},\begin{pmatrix}c\\ d\end{pmatrix},\begin{pmatrix}a+c\\ b+d\end{pmatrix}\Biggr{\}}\subseteq{\mathbb{Z}}^{2}}. In this paper, we study the spectral and non-spectral problems of μ M , 𝒟 {\mu_{M,{\mathcal{D}}}} . In this case that ( a b ) {(\begin{smallmatrix}a\\ b\end{smallmatrix})} and ( c d ) {(\begin{smallmatrix}c\\ d\end{smallmatrix})} are two independent vectors, we prove that if ρ - 1 ∈ ℤ {\rho^{-1}\in{\mathbb{Z}}} , then μ M , 𝒟 {\mu_{M,{\mathcal{D}}}} is a spectral measure if and only if ρ - 1 ∈ 2 ⁢ ℤ {\rho^{-1}\in 2{\mathbb{Z}}} . For the case that ( a b ) {(\begin{smallmatrix}a\\ b\end{smallmatrix})} and ( c d ) {(\begin{smallmatrix}c\\ d\end{smallmatrix})} are two dependent vectors, we first give the sufficient and necessary condition for L 2 ⁢ ( μ M , 𝒟 ) {L^{2}(\mu_{M,{\mathcal{D}}})} to contain an infinite orthogonal set of exponential functions. Based on this result, we can give the exact cardinality of orthogonal exponential functions in L 2 ⁢ ( μ M , 𝒟 ) {L^{2}(\mu_{M,{\mathcal{D}}})} when L 2 ⁢ ( μ M , 𝒟 ) {L^{2}(\mu_{M,{\mathcal{D}}})} does not admit any infinite orthogonal set of exponential functions by classifying the values of ρ.