Fuglede.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

4 declarations, 3 contain sorry, 0 declared as axioms

Fuglede's conjecture in dimensions 1 and 2

References:

Imports (1)

FormalConjecturesUtil

def Fuglede.FugledeConjectureFor Fuglede's conjecture in dimension n: A bounded subset of ℝ^n with positive Lebesgue measure is spectral iff it tiles ℝ^n by translation line 38 source
Source documentation
Fuglede's conjecture in dimension n: A bounded subset of ℝ^n with positive Lebesgue measure is spectral iff it tiles ℝ^n by translation.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Set, Fin, Real, Bornology.IsBounded, MeasurableSet, LT.lt, ENNReal, OfNat.ofNat, MeasureTheory.Measure, MeasureTheory.MeasureSpace.volume, Iff, isSpectral, tilesByTranslation
16 instances, coercions and auxiliary constants left out.
```
def FugledeConjectureFor (n : ℕ) : Prop :=
…
```
theorem Fuglede.FugledeConjecture.variants.dim_1 Fuglede's conjecture in one dimension: A bounded subset of ℝ with positive Lebesgue measure is spectral iff it tiles ℝ by translation sorry line 47 source
Source documentation
Fuglede's conjecture in one dimension: A bounded subset of ℝ with positive Lebesgue measure is spectral iff it tiles ℝ by translation.

depends on sorry what this means
Statement uses: Iff, True, Fuglede.FugledeConjectureFor, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem FugledeConjecture.variants.dim_1 :
    answer(sorry) ↔ FugledeConjectureFor 1 := by
…
```
Mentions: Fuglede.FugledeConjectureFor
theorem Fuglede.FugledeConjecture.variants.dim_2 Fuglede's conjecture in two dimensions: A bounded subset of ℝ^2 with positive Lebesgue measure is spectral iff it tiles ℝ^2 by translation sorry line 59 source
Source documentation
Fuglede's conjecture in two dimensions: A bounded subset of ℝ^2 with positive Lebesgue measure is spectral iff it tiles ℝ^2 by translation.
[Zh26], a July 2026 preprint, announces a negative answer in both directions, from two explicit $60$-point subsets of $\mathbb{Z}_{60} \times \mathbb{Z}_{12}$ lifted to finite unions of unit squares in $\mathbb{R}^2$. It is not yet published.

depends on sorry what this means
Statement uses: Iff, True, Fuglede.FugledeConjectureFor, OfNat.ofNat, Nat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem FugledeConjecture.variants.dim_2 :
    answer(sorry) ↔ FugledeConjectureFor 2 := by
…
```
Mentions: Fuglede.FugledeConjectureFor
theorem Fuglede.FugledeConjecture.variants.dim_3_or_higher Fuglede's conjecture fails in every dimension sorry line 71 source
Source documentation
Fuglede's conjecture fails in every dimension $n \geq 3$: for each such $n$ there is a bounded subset of $\mathbb{R}^n$ with positive Lebesgue measure that is spectral but does not tile $\mathbb{R}^n$ by translation, or that tiles $\mathbb{R}^n$ by translation but is not spectral. (Note that counterexamples in lower dimensions would also disprove the conjecture in higher dimensions.)

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Not, Fuglede.FugledeConjectureFor
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem FugledeConjecture.variants.dim_3_or_higher (n : ℕ) (hn : 3 ≤ n) :
    ¬ FugledeConjectureFor n := by
…
```
Mentions: Fuglede.FugledeConjectureFor