Hadamard triples generate self-affine spectral measures

Abstract:

Let $R$ be an expanding matrix with integer entries, and let $B,L$ be finite integer digit sets so that $(R,B,L)$ form a Hadamard triple on ${\mathbb {R}}^d$ in the sense that the matrix \begin{equation*} \frac {1}{\sqrt {|\det R|}}\left [e^{2\pi i \langle R^{-1}b,\ell \rangle }\right ]_{\ell \in L,b\in B} \end{equation*} is unitary. We prove that the associated fractal self-affine measure $\mu = \mu (R,B)$ obtained by an infinite convolution of atomic measures \begin{equation*} \mu (R,B) = \delta _{R^{-1} B}\ast \delta _{R^{-2}B}\ast \delta _{R^{-3}B}\ast \cdots \end{equation*} is a spectral measure, i.e., it admits an orthonormal basis of exponential functions in $L^2(\mu )$. This settles a long-standing conjecture proposed by Jorgensen and Pedersen and studied by many other authors. Moreover, we also show that if we relax the Hadamard triple condition to an almost-Parseval-frame condition, then we obtain a sufficient condition for a self-affine measure to admit Fourier frames.