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 ${\br}^d$ in the sense that the matrix $$ \frac{1}{\sqrt{|\det R|}}\left[e^{2\pi i \langle R^{-1}b,\ell\rangle}\right]_{\ell\in L,b\in B} $$ is unitary. We prove that the associated fractal self-affine measure $\mu = \mu(R,B)$ obtained by an infinite convolution of atomic measures $$ \mu(R,B) = \delta_{R^{-1} B}\ast\delta_{R^{-2}B}\ast\delta_{R^{-3}B}\ast... $$ 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.