Sufficient conditions for the spectrality of self-affine measures with prime determinant

Let mu(M,D) be a self-affine measure associated with an expanding matrix M and a finite digit set D. We study the spectrality of mu(M,D) when vertical bar det(M)vertical bar = vertical bar D vertical bar = p is a prime. We obtain several new sufficient conditions on M and D for mu(M,D) to be a spectral measure with lattice spectrum. As an application, we present some properties of the digit sets of integral self-affine tiles, which are connected with a conjecture of Lagarias and Wang.