In this paper, we study the spectral property of the self-affine measure [Formula: see text] generated by an expanding real matrix [Formula: see text] and the four-element digit set [Formula: see text]. We show that [Formula: see text] is a spectral measure, i.e. there exists a discrete set [Formula: see text] such that the collection of exponential functions [Formula: see text] forms an orthonormal basis for [Formula: see text], if and only if [Formula: see text] for some [Formula: see text]. A similar characterization for Bernoulli convolution is provided by Dai [X.-R. Dai, When does a Bernoulli convolution admit a spectrum? Adv. Math. 231(3) (2012) 1681–1693], over which [Formula: see text]. Furthermore, we provide an equivalent characterization for the maximal bi-zero set of [Formula: see text] by extending the concept of tree-mapping in [X.-R. Dai, X.-G. He and C. K. Lai, Spectral property of Cantor measures with consecutive digits, Adv. Math. 242 (2013) 187–208]. We also extend these results to the more general self-affine measures.

中文翻译：

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关于ℝ2上四位自仿射测量的光谱

在本文中，我们研究了由扩展实矩阵[公式：参见文本]和四元素数字集[公式：参见文本]生成的自仿射测度[公式：参见文本]的谱特性。我们证明[公式：见文本]是一个谱测量，即存在一个离散集[公式：见文本]，使得指数函数的集合[公式：见文本]形成[公式：见文本]的正交基, 当且仅当 [Formula: see text] 对于某些 [Formula: see text]。Dai [X.-R. 戴，伯努利卷积什么时候承认一个谱？进阶。数学。231(3) (2012) 1681–1693]，[公式：见正文]。此外，我们通过扩展 [X.-R. 戴，X.-G。He and CK Lai, Cantor Spectral property of Cantor measure with continuous digits, Adv. 数学。242 (2013) 187–208]。我们还将这些结果扩展到更一般的自仿射测量。