μM,D-Orthogonality and compatible pair

Let μM,D be a self-affine measure associated with an expanding integer matrix M ∈Mn(Z) and a finite subset D ⊂ Zn. In the present paper we study the μM,D-orthogonality and compatible pair conditions as well as relations between them. The research here is based on the structure of vanishing sums of roots of unity, and is closely related to the problem of spectral self-affine measure. © 2006 Elsevier Inc. All rights reserved.