The orthonormal wavelet implies wavelet-set conjecture

At least 4 years old · documented by 2021

Let $A$ be an invertible matrix, let $B$ be its transpose, and let $\Gamma$ be a full-rank lattice in $\mathbb{R}^n$ with dual lattice $\Gamma^*$ . An orthonormal wavelet for $(B,\Gamma^*)$ is an orthonormal wavelet associated with this dilation and translation pair, and an $(A,\Gamma)$ wavelet set is a measurable set whose integer $A$ -dilates and $\Gamma$ -translates each tile $\mathbb{R}^n$ measurably.

Orthonormal wavelet implies wavelet-set conjecture. For each pair $(B,\Gamma^*)$ such that there exists a $(B,\Gamma^*)$ orthonormal wavelet, there exists an $(A,\Gamma)$ wavelet set, where $B$ is the transpose of $A$ and $\Gamma^*$ is the dual lattice of $\Gamma$ .

This conjecture asks whether every pair admitting an orthonormal wavelet also admits a wavelet set. The paper notes that this implication holds in all cases currently known, while the general question remains open.

References

Primary source

Marcin Bownik and Darrin Speegle, “Simultaneous dilation and translation tilings of R^n”, arXiv:2109.10323 (2021).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

The conjecture remains open, while a reader-written submission offers only a conditional approach and does not prove the implication.

The conjecture asks whether every dilation and translation system admitting an orthonormal wavelet also admits a measurable wavelet set for the dual transposed system. It was attributed to Larson in the late 1990s and stated explicitly in 2021.

Known results

For $|\det A|>1$ , wavelet-set existence has a necessary-and-sufficient divergence criterion, but this does not establish the conjectured implication.
Positive cases include MRA wavelets and selected one-dimensional systems.
The stronger claim that every wavelet support contains a wavelet set remains open even in the one-dimensional dyadic case.
An expansive $A$ with any lattice $\Gamma$ guarantees existence of an $(A,\Gamma)$ wavelet set.

September 3, 2026 community submission

A submitted proof argues that, in a degree-two regime, measurable matching reduces the problem to a $\mathbb{Z}/2\mathbb{Z}$ cocycle being a measurable coboundary. It proves only a conditional wavelet-set theorem and explicitly leaves the general coboundary problem unresolved.

Current status (as of September 2026): The conjecture remains open; the only new development is an unverified conditional proof program, with no documented complete proof or counterexample.

Sources

ProofFROM ORTHONORMAL WAVELETS TO WAVELET SETS Incidence Graphs, Measurable Markers, and a Z/2Z-Cocycle Formulation Expanded journal-style research manuscript Prepared as a rigorous proof program; unresolved implications are explicitly identified.See full solutionHide full solution

We study the implication from the existence of an orthonormal wavelet to the existence of a measurable wavelet set for a dual dilation–translation pair. Let A be an invertible matrix, B=A^T, Γ a full-rank lattice, and Γ* its dual lattice. The central question is whether the existence of an orthonormal (B,Γ*) wavelet forces the existence of an (A,Γ) wavelet set. We develop a structural reduction centered on the Fourier support of an orthonormal wavelet. In the degree-two regime, the relevant simultaneous translation–dilation selection problem can be encoded by a measurable bipartite incidence graph of maximum degree two. The combinatorial matching problem is then elementary: every bi-infinite component has two alternating perfect matchings. The difficulty is measurable selection of one of these alternatives on almost every component. We analyze several candidate selection mechanisms. A nearest-to-zero selector fails because typical affine orbits can have infimum zero without attaining zero. A measurable first-return construction for the doubling map supplies coherent local markers and a measurable parity transport along dilation chains. Passing to the fractional part makes these markers invariant under integer translations. The remaining compatibility is naturally expressed as a measurable Z/2Z cocycle and, in the desired case, as a measurable coboundary. We prove a conditional wavelet-set theorem: if the relevant incidence cocycle admits such a measurable coboundary, then the Fourier support contains a measurable wavelet set. We do not claim that the remaining coboundary problem is solved in full generality. Instead, we isolate it as the precise measurable obstruction and formulate a concrete research program for resolving it.

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Orthonormal_Wavelet_to_Wavelet_Set_Research_Report_rev.1-compressed.pdf760,077 bytesOpen