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Hadamard triples generate self-affine spectral measures
Dorin Ervin Dutkay, Chun-Kit Lai, John Haussermann
TL;DR
The paper addresses whether self-affine measures associated with Hadamard triples always admit orthonormal exponential bases. It uses Fourier-transform zero sets, invariant-subspace analysis, lattice reductions, and quasi-product structure to prove that they do, settling the Jorgensen–Pedersen conjecture.
Problem
Jorgensen and Pedersen asked which measures admit orthonormal bases of exponential functions, and it remained conjectured that all Hadamard triples generate spectral self-affine measures.
Method
The proof analyzes the Fourier-transform zero set dynamically, establishes a nontrivial invariant rational subspace, and conjugates the system into quasi-product form.
Results
Every self-affine measure associated with a Hadamard triple is spectral.
Takeaways & Limitations
The result completes the conjectured Hadamard-triple construction of spectral self-affine measures beyond previously treated special cases.
Takeaways & Limitations
The theorem treats equal-weight self-affine measures; the paper states that unequal weights cannot admit a spectrum under the cited no-overlap result.
Abstract
from arXiv · showhide
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.
1. Introduction
The paper studies when self-affine measures generated by Hadamard triples admit orthonormal exponential bases, resolving the Jorgensen–Pedersen conjecture affirmatively. It defines the relevant spectral and Hadamard-triple frameworks and places the result among earlier partial cases and related constructions.
- A spectral measure is a compactly supported probability measure whose L2 space has an orthonormal basis of exponential functions.
- A Hadamard triple consists of an expansive integer matrix R and equally sized finite digit sets B and L for which the associated matrix is unitary.
- Hadamard triples generate equal-weight self-affine measures through an affine iterated function system and its invariant probability measure.
- The paper proves that every self-affine measure associated with a Hadamard triple is spectral, establishing the conjecture proposed by Jorgensen and Pedersen.
- Earlier work established the theorem in dimension 1 and under additional reducibility or structural assumptions in higher dimensions.
2. Preliminaries
The preliminaries develop lattice reductions, Fourier-transform identities, and invariant-zero-set dynamics used to prove spectrality. Conjugation and quasi-product reductions simplify the Hadamard triple while preserving the relevant spectral property.
- Conjugating a Hadamard triple preserves the Hadamard property, while spectrality transfers with the spectrum transformed by (M^T)^-1.
- The lattice generated by the digit expansions can be reduced to full rank or to a lower-dimensional conjugate triple without loss of generality.
- The Fourier transform of the self-affine measure satisfies an iterative scaling identity involving the mask function u_B.
- The proof rules out the trivial invariant subspace W={0}, using the full-rank lattice assumption to derive a contradiction with bµ(0)=1.
- A nontrivial invariant subspace enables an integer conjugation to a more regular quasi-product structure, with component triples that are themselves Hadamard triples.
3. The quasi-product form
When the zero set Z is nonempty, the Hadamard triple can be conjugated into a quasi-product form, decomposing the digit structure into lower-dimensional Hadamard triples.
- Quasi-product structure: The section proves that a Hadamard triple with Z ≠ ∅ is conjugate to a quasi-product form.The proof analyzes Z as an invariant dynamical-system set and uses integer conjugation.
- Lattice structure: The matrix Q is integer, satisfies |det Q| ≥ 2, and intertwines R2 with another integer matrix through R2Q = Q R̃2.This arises because the relevant full-rank lattice is a proper sublattice.
- Lower-dimensional triples: For every b1, B2(b1) is a complete representative set modulo R2(Z^(d−r)), yielding a Hadamard triple (R2, B2(b1), π2(L)).The corresponding cardinality is |det R2| = #π2(L).
- Lower-dimensional triples: For every ℓ2, (R1, π1(B), L1(ℓ2)) is also a Hadamard triple, with #L1(ℓ2) = #π1(B) = N/|det R2|.The two lower-dimensional systems inherit the Hadamard structure after choosing a suitable representative L.
4. Proof of the theorem
The proof reduces the self-affine measure to a quasi-product structure and constructs spectra for its component measures. An induction argument then combines these component spectra to obtain a spectrum for the original measure.
- Quasi-product structure: The measure is represented through a first-coordinate attractor and conditional second-coordinate measures built from digit sequences.Almost every first-coordinate point corresponds uniquely, up to measure zero, to a digit sequence; product probabilities induce the component measures.
- Component spectra: A lattice Γ2 is constructed so that it is a spectrum for the second component for µ1-almost every first-coordinate point.The construction passes through a modified pair whose attractor tiles by Z × ˜Γ2; Fuglede's result then gives a spectrum of the form Z × Γ2.
- Inductive step: The first component is spectral by induction, while the second-component lattice spectrum holds almost everywhere in the first coordinate.The proof uses the lower-dimensional Hadamard triple and Proposition 4.4 to obtain Λ1 and Γ2.
- Quasi-product structure: When the invariant set is nonempty, conjugation puts (R, B) into quasi-product form, which is the structural setting for the proof.The proof first handles the empty invariant-set case separately, then applies the quasi-product reduction.
- Inductive step: The product set Λ1 × Γ2 is a spectrum for µ, completing the induction and the proof of the theorem.The final combination follows from the decomposition of the measure and the cited spectral lemmas.