Spectral self-affine measures with prime determinant

The self-affine measure $mu _{M,D}$ relating to an expanding matrix $Min M_{n}(mathbb Z )$ and a finite digit set $Dsubset mathbb Z ^n$ is a unique probability measure satisfying the self-affine identity with equal weight. In the present paper, we shall study the spectrality of $mu _{M,D}$ in the case when $|det (M)|=p$ is a prime. The main result shows that under certain mild conditions, if there are two points $s_{1}, s_{2}in mathbb R ^{n}, s_{1}-s_{2}in mathbb Z ^{n}$ such that the exponential functions $e_{s_{1}}(x), e_{s_{2}}(x)$ are orthogonal in $L^{2}(mu _{M,D})$ , then the self-affine measure $mu _{M,D}$ is a spectral measure with lattice spectrum. This gives some sufficient conditions for a self-affine measure to be a lattice spectral measure.