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 ${\br}^d$. We prove that the associated self-affine measure $μ= μ(R,B)$ is a spectral measure, which means it admits an orthonormal bases of exponential functions in $L^2(μ)$. This settles a long-standing conjecture proposed by Jorgensen and Pedersen and studied by many other authors.