The cardinality of orthogonal exponentials of planar self-affine measures with two-element digit set

Abstract

Abstract Let μ M , D {{\mu_{M,D}}} be the planar self-affine measure determined by an expanding integer matrix M ∈ M 2 ⁢ ( ℤ ) {M\in M_{2}(\mathbb{Z})} and a two-element digit set D ⊂ ℤ 2 {D\subset\mathbb{Z}^{2}} . It has been shown that the spectral or non-spectral problem on μ M , D {\mu_{M,D}} is only related to trace ⁡ ( M ) := r 1 {\operatorname{trace}(M):=r_{1}} and det ⁡ ( M ) := r 2 {\det(M):=r_{2}} . In the case when M ∈ M 2 ⁢ ( ℤ ) {M\in M_{2}(\mathbb{Z})} is an expanding matrix and r 1 2 = 3 ⁢ r 2 {r_{1}^{2}=3{r_{2}}} , r 2 ∈ 2 ⁢ ℤ + 1 ∖ { ± 1 } {{r_{2}}\in 2\mathbb{Z}+1\setminus\{\pm 1\}} , the Hilbert space L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} contains at most a finite number of orthogonal exponentials, and μ M , D {{\mu_{M,D}}} is a non-spectral measure. The remaining problem in this case is to determine the best upper bound on the cardinality of orthogonal exponentials in the Hilbert space L 2 ⁢ ( μ M , D ) {L^{2}(\mu_{M,D})} . In the present paper, we further the above research to show that there are at most 16 mutually orthogonal exponentials in the corresponding Hilbert space, and the number 16 is the best upper bound. This completes the non-spectrality of self-affine measures in the above case.