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Conjecture

Fuglede's Conjecture

Analysis

Fuglede's conjecture, proposed by Bent Fuglede in 1974, states that a domain of positive finite Lebesgue measure in Euclidean space is a spectral set if and only if it tiles space by translation. Terence Tao resolved the conjecture in the negative for most dimensions in 2004. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/

Facts
Statement
Every domain of R^d, meaning a subset of R^d with positive finite Lebesgue measure, is a spectral set if and only if it tiles R^d by translation. 1
Proposed Year
1974 1
Progress Toward Resolution
Resolved in the negative for most dimensions: Tao showed in 2004 it is false for d of 5 or more, and it was later shown false for d = 3 and 4; it remains unknown for d = 1 and 2. Lev and Matolcsi settled it affirmatively for convex domains in all dimensions in 2019. 1
Classification
Resolution Status
Disproven 1
Chronology
Resolved Year
2004 1
Prize Status
Prize Status (category)
No Prize Offered 1
Sources
1. Fuglede's Conjecture (Wikipedia)
Wikimedia Foundation
  • Lead section
    proposed by Bent Fuglede in 1974
  • Lead section, second sentence
    is a spectral set if and only if it tiles
  • Partial results section
    In 2019, Nir Lev and Máté Matolcsi settled the conjecture for convex domains affirmatively in all dimensions.
View the Source
Fuglede's conjecture (Wikipedia)
Wikipedia Fuglede's conjecture lead paragraph (w-bbfill-psymath4-0926)
Quote, Wikipedia Fuglede's conjecture lead paragraph (w-bbfill-psymath4-0926)
resolved in the negative for most dimensions by Terence Tao in 2004
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