Arbitrarily Sparse Spectra for Self-Affine Spectral Measures

Given an expansive matrix R ∈ M d (ℤ) and a finite set of digit B taken from ℤ d / R ( ℤ d ). 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.