Arbitrarily Sparse Spectra for Self-Affine Spectral Measures

摘要

Given an expansive matrix R is an element of M-d(DOUBLE-STRUCK CAPITAL Z) and a finite set of digit B taken from DOUBLE-STRUCK CAPITAL Z(d)/R(DOUBLE-STRUCK CAPITAL 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 ?, 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 ? in the sense that its Beurling dimension is zero.

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