A class of spectral self-affine measures with four-element digit sets

Abstract

The self-affine measure μM,D associated with an expanding matrix M∈Mn(Z) and a finite digit set D⊂Zn is uniquely determined by the self-affine identity with equal weight. In this paper we construct a class of self-affine measures μM,D with four-element digit sets in the higher dimensions (n≥3) such that the Hilbert space L2(μM,D) possesses an orthogonal exponential basis. That is, μM,D is spectral. Such a spectral measure cannot be obtained from the condition of compatible pair. This extends the corresponding result in the plane.