论文标题
自我伴随光谱测量的任意稀疏光谱
Arbitrarily sparse spectra for self-affine spectral measures
论文作者
论文摘要
给定M_D（{\ Mathbb Z}）中的扩展矩阵$ r \，以及从$ {\ Mathbb z}^d/r（{\ Mathbb z}^d）$中取的一组有限的数字$ b $。先前显示的是，如果我们可以找到$ L $，以便$（r，b，l）$形成A Hadamard三重，那么由$（r，b）$产生的相关分形自动措施承认了某些频率集$λ$的指数正顺数基础，因此被称为光谱量度。在本文中，我们表明，如果＃$ b <| \ det（r）| $，不仅是光谱，我们还可以任意构建稀疏频谱$λ$，从而使其beurling dimension为零。
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 $Λ$, 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.
