Nonspectrality of Certain Self-Affine Measures on ℝ3

Abstract

We will determine the nonspectrality of self-affine measure ${\mu }_{B,D}$ corresponding to $B=\mathrm{diag}[p_1,p_2,p_3]\text{(}{p}_1\in (2\mathbb{Z}+1)\setminus \{\pm 1\}$, $p_2\in 2\mathbb{Z}\setminus \{0\})$, and $D=\{0,e_1,e_2,e_3\}$ in the space ${ℝ}^3$ is supported on $T(B,D)$, where $e_1,e_2$, and $e_3$ are the standard basis of unit column vectors in ${ℝ}^3$, and there exist at most $4$ mutually orthogonal exponential functions in $L^2({\mu }_{B,D})$, where the number $4$ is the best. This generalizes the known results on the spectrality of self-affine measures.