On the spectra of self-affine measures with three digits (Q2317452)

The authors study probability measures $\mu$ with compact support in $\mathbb{R}^d$ with an extra property: there exists a discrete set $\Lambda\subset\mathbb{R}^d$ such that $E_{\lambda}=\{\exp{(2\pi i \langle\xi,\lambda\rangle)}:\lambda\in\Lambda\}$ forms an orthonormal basis of $L^2(\mu)$. \par The main result is: \par Theorem 1.1. Assume that $B\equiv 0 \pmod{3}$ and $T$ contains $\text{0}=(0,0)^t$ as an interior point. Then $E_{\Lambda}$ with $0 \in\Lambda$ is a maximal orthogonal family of $L^2(\mu)$ if and only if there exist $C_{j_1,\ldots,j_n}=\{\alpha_{j_1\ldots j_n i}:i=1,2,3\}\subset\mathcal{M}\ (n=0,1,2,\ldots)$ and non-empty sets $\Lambda_{j_1\ldots j_n}\subset\mathbb{Z}^2$ satisfying the following:\par (i) There exist $j_1,j_2,\ldots$ such that $\alpha_{j_1\ldots j_n}=0$ for all $n>0$;\par (ii)$\frac{1}{3}BC_{j_1\ldots j_n}$ are compatible sets for all $n\geq 0$ and $j_1,\ldots, j_n\in\{1,2,3\}$;\par (iii) $\Lambda$ can be written in the form $$\Lambda=\bigcup\Bigl\{\sum_{n=1}^k\,\frac{1}{3}B^n\alpha_{j_1\ldots j_n}+\frac{1}{3}B^{k+1}\Lambda_{j_1\ldots j_n}: j_1,\ldots, j_k\in \{1,2,3\}\Bigr\}$$ for all $k\in\mathbb{N}$.\par Furthermore, the sets $E_{\frac{1}{3}B\Lambda_{j_1\ldots j_n}}$ are also maximal families in $L^2 (\mu)$ for all $n>0$ and $j_1,\ldots, j_n\in\{1,2,3\}$.\par The other result is:\par Theorem 1.3. Under the assumptions of Theorem 1.2, let $E_{\Lambda}$ be a maximal orthogonal family of $L^2(\mu)$ such that $0\in\Lambda$ and let the tree $\mathcal{T}(\Lambda)$ be defined ``as above'', while $0<\delta\), \(\rho<1$ follow from Lemma 2.6 and Lemma 2.7, then \par (i) If there exists an integer sequence $\{n_k\}_k$ such that $n_{k+1}-M_{n_k}\rightarrow\infty$ as $k\rightarrow\infty$ and $\sum_{k=1}^{\infty}\,\delta^{N_{n_k,n_{k+1}}}=\infty$, then $\Lambda$ is a spectrum of $\mu$. \par (ii) If $\sum_{k=1}^{\infty}\,\rho^{L_k}<\infty$, then $\Lambda$ is not a spectrum of $\mu$.\par Here: \par (a) if $E_{\Lambda}$ is an orthogonal family of $L^2(\mu)$ and $E_{\Lambda\cup\Gamma}$ is it not for any subset $\Gamma\not\subseteq\Lambda$, $E_{\Lambda}$ is called maximal.\par (b)$A$ is an integral expansive $2\time 2$ matrix (the absolute values of the eigenvalues of $A$ are larger than one), $B=A^t$ its transpose. \par (c) $\mathcal{M}=\text{Z}^2\cap\{B(x,y)^t: -\frac{1}{2}<x,y\leq\frac{1}{2}\}$\par (d) $T$ is the unique non-empty compact subset of $\text{R}^2$ with $BT=\cup{c\in\mathcal{M}}\,(T+c)$.\par (e) If $C=\{s_0,s_1,s_2\}\subset\text{R}^2$ with three elements and $D=\{e_0,e_1,e_2\}$ ($e_0=(0,0)^t, e_1=(1,0)^t, e_2=(0,1)^t$), then $C$ is a compatible set if $$H_{A^{-1}D,C}=[\frac{1}{\sqrt{3}}\exp{(2\pi i<A^{-1}e_j,s_k>)}]_{0\leq j, k\leq 2}$$ is a unitary matrix.\par (f) ``as above'' indicates the location in the paper where the definition is given.\par (g) The lemma's 2.6 and 2.7 are given in \S 2.