Arbitrarily sparse spectra for self-affine spectral measures

Given an expansive matrix $R\in M_d(\mathbb{Z})$ and a finite set of digit $B$ taken from ${\mathbb{Z}}^d\mathrm{/}R({\mathbb{Z}}^d)$. 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 $\mathrm{\Lambda }$, and hence it is termed as a spectral measure. In this paper, we show that if #$B<|\det ®|$, not only it is spectral, we can also construct arbitrarily sparse spectrum $\mathrm{\Lambda }$ in the sense that its Beurling dimension is zero.