SpectralSets.lean`

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8 declarations, 0 contain sorry, 0 declared as axioms

Spectral sets

This file defines the basic notions used in problems related to spectral sets and Fuglede's conjecture: translations, exponential characters, spectral pairs, spectral sets, and translational tilings in Euclidean space.

Imports (2)

Imported by 1 file

FormalConjecturesForMathlib.lean

def translateSet The translate of a set Ω by a vector t line 35 source
Source documentation
The translate of a set Ω by a vector t.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set, Fin, Real
Proof uses: Set.ofPred, Membership.mem, HSub.hSub
4 instances, coercions and auxiliary constants left out.
```
def translateSet {d : ℕ} (Ω: Set (Fin d → ℝ)) (t : Fin d → ℝ) :
    Set (Fin d → ℝ) :=
…
```
def exponentialCharacter The exponential character x ↦ exp(2π i ⟪ξ, x⟫) line 40 source
Source documentation
The exponential character x ↦ exp(2π i ⟪ξ, x⟫).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Real, Complex
Proof uses: Complex.exp, HMul.hMul, OfNat.ofNat, Real.pi, Complex.I, Finset.sum, Finset.univ
9 instances, coercions and auxiliary constants left out.
```
noncomputable def exponentialCharacter {d : ℕ} (ξ x : Fin d → ℝ) : ℂ :=
…
```
theorem continuous_exponentialCharacter The exponential character is continuous line 44 source
Source documentation
The exponential character is continuous.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Real, Continuous, Complex, exponentialCharacter
Proof uses: id, Eq.mpr, Complex.exp, HMul.hMul, OfNat.ofNat, Real.pi, Complex.I, Finset.sum, Finset.univ, Eq, congrArg, funext, Eq.trans, Complex.ofReal_sum, Finset.sum_congr, Eq.refl, Finset, Membership.mem, Complex.ofReal_mul, Continuous.comp' and 5 more
43 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem continuous_exponentialCharacter {d : ℕ} (ξ : Fin d → ℝ) :
    Continuous (exponentialCharacter ξ) := by
…
```
Mentions: exponentialCharacter
theorem norm_exponentialCharacter The exponential character has norm one line 50 source
Source documentation
The exponential character has norm one.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Real, Eq, Norm.norm, Complex, exponentialCharacter, OfNat.ofNat
Proof uses: id, HMul.hMul, Real.pi, Complex.I, Finset.sum, Finset.univ, Eq.mpr, congr, congrArg, Eq.trans, Complex.ofReal_sum, Finset.sum_congr, Eq.refl, Finset, Membership.mem, Complex.ofReal_mul, Mathlib.Tactic.Ring.of_eq, HAdd.hAdd, HPow.hPow, Nat.rawCast and 19 more
42 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
```
theorem norm_exponentialCharacter {d : ℕ} (ξ x : Fin d → ℝ) :
    ‖exponentialCharacter ξ x‖ = 1 := by
…
```
Mentions: exponentialCharacter
theorem exponentialCharacter_memLp On a finite-measure set, every exponential character belongs to L^2 line 61 source
Source documentation
On a finite-measure set, every exponential character belongs to L^2.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set, Fin, Real, Ne, ENNReal, MeasureTheory.Measure, MeasureTheory.MeasureSpace.volume, Top.top, MeasureTheory.MemLp, Complex, exponentialCharacter, OfNat.ofNat, MeasureTheory.Measure.restrict
Proof uses: Eq.mpr, MeasureTheory.Integrable, HPow.hPow, Norm.norm, id, Eq, congrArg, propext, MeasureTheory.memLp_two_iff_integrable_sq_norm, Continuous.aestronglyMeasurable, continuous_exponentialCharacter, congrFun', autoParam, funext, norm_exponentialCharacter, MeasureTheory.IntegrableOn, LT.lt, Eq.trans, Or, ENorm.enorm and 11 more
110 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem exponentialCharacter_memLp {d : ℕ} {Ω : Set (Fin d → ℝ)}
    (hΩ_finite : volume Ω ≠ ⊤) (ξ : Fin d → ℝ) :
    MemLp (exponentialCharacter ξ) 2 (volume.restrict Ω) := by
…
```
Mentions: exponentialCharacter
def spectralPair Documented declaration: spectralPair line 78 source
Source documentation
(Ω, Λ) is called a spectral pair if Ω is a (Lebesgue) measurable set of positive finite measure and Λ is a spectrum for Ω, i.e. a set Λ of frequencies such that the associated exponential characters form an orthogonal basis of L^2(Ω).
The positivity requirement matters: if Ω is a null set then L^2(Ω) is the zero space and every Λ would vacuously be a spectrum.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set, Fin, Real
Proof uses: And, MeasureTheory.NullMeasurableSet, MeasureTheory.MeasureSpace.volume, LT.lt, ENNReal, OfNat.ofNat, MeasureTheory.Measure, Exists, Ne, Top.top, Subtype, MeasureTheory.AEEqFun, Complex, MeasureTheory.Measure.restrict, Membership.mem, AddSubgroup, MeasureTheory.Lp, MeasureTheory.MemLp.toLp, exponentialCharacter, exponentialCharacter_memLp and 6 more
51 instances, coercions and auxiliary constants left out.
```
def spectralPair {d : ℕ} (Ω Λ : Set (Fin d → ℝ)) : Prop :=
…
```
def isSpectral A set is spectral if it belongs to a spectral pair line 89 source
Source documentation
A set is spectral if it belongs to a spectral pair.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set, Fin, Real
Proof uses: Exists, spectralPair
```
def isSpectral {d : ℕ} (Ω : Set (Fin d → ℝ)) : Prop :=
…
```
def tilesByTranslation A set Ω tiles ℝ^d by translation if there exists a set T of translation vectors such that the translates of Ω by T cover ℝ^d without overlap line 96 source
Source documentation
A set Ω tiles ℝ^d by translation if there exists a set T of translation vectors such that the translates of Ω by T cover ℝ^d without overlap.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Set, Fin, Real
Proof uses: Exists, And, Set.Countable, Eq, ENNReal, MeasureTheory.Measure, MeasureTheory.MeasureSpace.volume, SDiff.sdiff, Set.univ, Set.iUnion, Membership.mem, translateSet, OfNat.ofNat, Ne, Inter.inter
11 instances, coercions and auxiliary constants left out.
```
def tilesByTranslation {d : ℕ} (Ω : Set (Fin d → ℝ)) : Prop :=
…
```