A Characterization on the Spectra of Self-Affine Measures

A discrete set Lambda subset of R-d is called a spectrum for the probability measure mu if the family of functions {e(2 pi i <lambda,) (x >) : lambda is an element of Lambda} forms an orthonormal basis for the Hilbert space L-2( mu). In this paper, we will give a characterization of the spectra of self- affine measures generated by compatible pairs in R-d. As an application, we show, for the Cantor measure mu(b, q) on R with consecutive digit set and any integer p is an element of Z with gcd( p, q) = 1, that the set {Lambda subset of R : Lambda and p Lambda are both spectra for mu(b, q) and 0 is an element of Lambda} has the cardinality of the continuum.