Breakthrough Tracker record
Both directions of Fuglede’s conjecture fail in dimension two
The paper gives two explicit 60-point constructions in a finite abelian group and transfers them to bounded subsets of the plane. One planar set tiles by translations but is not spectral; the other is spectral but does not tile.
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- Stable ID
math-fuglede-dimension-two-counterexamples-2026- Revision
math-fuglede-dimension-two-counterexamples-2026.v1- Field
- Mathematics · Harmonic analysis and combinatorics
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Provisional preprint claim
- Last checked
AI role
No AI role was disclosed in the inspected manuscript.
Record details
- Problem or result
- Fuglede’s conjecture in the two-dimensional unrestricted measurable setting
- Authors
- Tao Zhang
- Institutions
- Xidian University
- Result date
- Preprint submitted July 17, 2026
Why it matters
Counterexamples were already known in dimensions three and higher, while the unrestricted two-dimensional case had remained unresolved.
Limits
This is a first-version preprint without located independent specialist assessment or peer review. The result concerns unrestricted measurable sets in the plane; narrower versions of Fuglede’s problem can remain open.
Sources
Correction and revision history
- 2026-07-23 — Prepared as a provisional tracker addition after checking the explicit constructions and the stated finite-to-infinite transfer; independent assessment remains pending.
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