Arbitrarily Sparse Spectra for Self-Affine Spectral Measures

Given an expansive matrix R ∈ Md(ℤ) and a finite set of digit B taken from ℤd/R(ℤd). It was shown previously that if we can find an L such that (R, B, L) forms a Hadamard triple, then the associated fractal self-affine measure generated by (R, B) admits an exponential orthonormal basis of certain frequency set Λ, and hence it is termed as a spectral measure. In this paper, we show that if #B < ∣det(R)∣, not only it is spectral, we can also construct arbitrarily sparse spectrum Λ in the sense that its Beurling dimension is zero.