Non-spectral problem for a class of planar self-affine measures
Given an \(n \times n\) expanding matrix \(M \in M_n(\mathbb R)\) and a finite subset \(D \subset \mathbb R^n\) with cardinality \(|D|\), the corresponding self-affine measure \(\mu_{M, D}\) is the unique probability measure \(\mu := \mu_{M, D}\) which satisfies the equation \[ \mu = \frac 1 {|D|} \sum_{d \in D} \mu \circ \phi_d^{-1}, \] where \(\phi_d(x) = M^{-1}(x + d)\) \((x \in \mathbb R^n)\). \(\mu_{M, D}\) is supported on the attractor of the iterated function system \(\{\phi_d, d \in D\}\). This paper under review is concerned with the spectral and non-spectral problems on \(\mu_{M, D}\). Such problems have attracted much attention in recent years due to their connections to harmonic analysis, dynamical systems, fractal geometry and number theory. Specifically, the author considers the case that \[ M = \left[ \begin{matrix} a \;&b \\ 0 \;&c \end{matrix} \right] \quad \text{ and } \quad D = \left\{ \binom{0}{0}, \binom{1}{0}, \binom{0}{1}\right\} \] and proves that if \(a, b, c \in {\mathbb Z}\), \(|a| > 1, |c| > 1\) and \(ac \in {\mathbb Z} \backslash (3 {\mathbb Z})\), then there exist at most 3 mutually orthogonal exponentials in \(L^2(\mu_{M, D})\) and the number 3 is the best. The proof of the above theorem depends on the characterization of the zero set \(Z(\hat{\mu}_{M, D})\) of the Fourier transform of \(\mu_{M, D}\).
scientific article; zbMATH DE number 5480704
scientific article