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 L such (R, B, L) forms Hadamard triple, then the associated fractal self-affine measure generated by B) admits exponential orthonormal basis certain frequency Λ, hence it is termed as spectral measure. In this paper, show #B < ∣det(R)∣, not only spectral, also construct arbitrarily sparse spectrum Λ in sense its Beurling dimension zero.