Hadamard triples generate self-affine spectral measures

TL;DR

This paper proves that self-affine measures generated by Hadamard triples are spectral, confirming a long-standing conjecture and establishing the existence of orthonormal bases of exponential functions in their L^2 spaces.

Contribution

It demonstrates that all self-affine measures from Hadamard triples are spectral, resolving a major open problem in the field.

Findings

Self-affine measures from Hadamard triples are spectral.

Existence of orthonormal bases of exponential functions in these measures.

Resolution of a long-standing conjecture by Jorgensen and Pedersen.

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 $\mu = \mu(R,B)$ is a spectral measure, which means it admits an orthonormal bases of exponential functions in $L^2(\mu)$. This settles a long-standing conjecture proposed by Jorgensen and Pedersen and studied by many other authors.