Spectral Property of Self-Affine Measures on Rn

For any integer m > 1, let D subset of Z(n) be a finite digit set such that Z(mD) = boolean OR(k)(i=1) Z(i) for some finite integer k, (Z(i)-Z(i))\Z(n) subset of Z(i) subset of (m(-1)Z\Z)(n) and Z(i) not subset of (m'(-1) Z\Z)(n) for all 0 < m' < m, where Z(m(D)) = {x : Sigma(d is an element of D) e(2 pi,i(d,x)) = 0}. Let M = diag[b(1),.....,b(n)] be a real expansive diagonal matrix and mu(M,D) be the self-affine measure on Rn defined by mu M, D(.) = 1/vertical bar D vertical bar Sigma(d is an element of D) mu(M,D) (M(.) - d). In this paper, we first give the sufficient and necessary condition for L-2(mu(M,D)) to contain an infinite orthogonal exponentials for any integer m > 1. Furthermore, we show that, if m is a prime, mu(M,D) is a spectral measure if and only if m vertical bar b(i), i = 1, 2,..., n. This extends known results in [5,6,28].