Arbitrarily sparse spectra for self-affine spectral measures

Abstract: Given an expansive matrix $R\in M_d({\mathbb Z})$ and a finite set of digit $B$ taken from $ {\mathbb Z}^d/R({\mathbb 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 $\Lambda$, 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 $\Lambda$ in the sense that its Beurling dimension is zero.