Non-spectrality of planar self-affine measures with three-elements digit set
Given an \(n\times n\) expanding real matrix \(M\) and a finite subset \(D\subset {\mathbb R}^n\), there exist a finite number of contractive affine maps \(\phi_d(x) = M^{-1}(x + d)\) (\(x\in {\mathbb R}^n\), \(d\in D\)) and a measure \(\mu_{M, D}\), called self-affine measure. The self-affine measure \(\mu_{M, D}\) is a spectral measure if there exists a discrete set \(\Lambda \subset {\mathbb R}^n\) such that the exponential function system \(E_{\Lambda} = \{e^{2\pi i \langle\lambda, \cdot\rangle} : \lambda\in\Lambda\}\) forms an orthogonal basis for \(L^2(\mu_{M, D})\). The paper concerns the question: under what conditions on \(M\) and \(D\), is \(\mu_{M, D}\) a spectral or non-spectral measure? In [J. Funct. Anal. 255, No. 11, 3125--3148 (2008; Zbl 1211.28007)], the author conjectured that for an expanding integer matrix \(M\in M_n({\mathbb Z})\) and a finite digit set \(D\subset \mathbb Z^n\), if the cardinality \(|D| \not\in W(|\det(M)|)\), then \(\mu_{M, D}\) is a non-spectral measure, and there are at most finite number of orthogonal exponents in \(L^2(\mu_{M, D})\). Here for each non-negative integer \(m\), \(W(m)\) denotes the set ofnon-negative integer combinations of \(p_1, p_2, \ldots, p_r\), where \(m = p_1^{b_1} p_2^{b_2} \cdots p_r^{b_r}\) is the prime factorization of \(m\). In this paper, the author proves that this conjecture is true in the plane \(\mathbb R^2\) for some particular three-elements digit set \(D\).
scientific article; zbMATH DE number 5577363
scientific article