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module
public import Mathlib.MeasureTheory.Function.L2Space
public import Mathlib.MeasureTheory.Measure.Lebesgue.Basic@[expose] public sectionSpectral 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.
open Real Set Complex MeasureTheoryopen scoped InnerProductSpace
The translate of a set Ω by a vector t.
def translateSet {d : ℕ} (Ω: Set (Fin d → ℝ)) (t : Fin d → ℝ) :
Set (Fin d → ℝ) :=
{x | x - t ∈ Ω}
The exponential character x ↦ exp(2π i ⟪ξ, x⟫).
noncomputable def exponentialCharacter {d : ℕ} (ξ x : Fin d → ℝ) : ℂ :=
Complex.exp (2 * π * Complex.I * (∑ i, ξ i * x i))The exponential character is continuous.
theorem continuous_exponentialCharacter {d : ℕ} (ξ : Fin d → ℝ) :
Continuous (exponentialCharacter ξ) := d:ℕξ:Fin d → ℝ⊢ Continuous (exponentialCharacter ξ)
d:ℕξ:Fin d → ℝ⊢ Continuous fun x ↦ cexp (2 * ↑π * I * ↑(∑ i, ξ i * x i))
All goals completed! 🐙The exponential character has norm one.
All goals completed! 🐙
On a finite-measure set, every exponential character belongs to L^2.
d:ℕΩ:Set (Fin d → ℝ)hΩ_finite:volume Ω ≠ ⊤ξ:Fin d → ℝ⊢ Integrable (fun x ↦ ‖exponentialCharacter ξ x‖ ^ 2) (volume.restrict Ω)
simp_rw d:ℕΩ:Set (Fin d → ℝ)hΩ_finite:volume Ω ≠ ⊤ξ:Fin d → ℝ⊢ Integrable (fun x ↦ ‖exponentialCharacter ξ x‖ ^ 2) (volume.restrict Ω)d:ℕΩ:Set (Fin d → ℝ)hΩ_finite:volume Ω ≠ ⊤ξ:Fin d → ℝ⊢ Integrable (fun x ↦ 1 ^ 2) (volume.restrict Ω)]
d:ℕΩ:Set (Fin d → ℝ)hΩ_finite:volume Ω ≠ ⊤ξ:Fin d → ℝ⊢ IntegrableOn (fun x ↦ 1 ^ 2) Ω volume
All goals completed! 🐙(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.
def spectralPair {d : ℕ} (Ω Λ : Set (Fin d → ℝ)) : Prop :=
NullMeasurableSet Ω volume ∧ 0 < volume Ω ∧
∃ hΩ_finite : volume Ω ≠ ⊤,
let e : (Fin d → ℝ) → MeasureTheory.Lp ℂ 2 (volume.restrict Ω) := fun ξ ↦
MeasureTheory.MemLp.toLp (exponentialCharacter ξ) (exponentialCharacter_memLp hΩ_finite ξ)
(∀ ξ₁ ∈ Λ, ∀ ξ₂ ∈ Λ, ξ₁ ≠ ξ₂ → ⟪e ξ₁, e ξ₂⟫_ℂ = 0) ∧
Submodule.topologicalClosure (Submodule.span ℂ (Set.range fun (ξ : Λ) ↦ e ξ.1)) = ⊤A set is spectral if it belongs to a spectral pair.
def isSpectral {d : ℕ} (Ω : Set (Fin d → ℝ)) : Prop :=
∃ Λ : Set (Fin d → ℝ), spectralPair Ω ΛA 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.
def tilesByTranslation {d : ℕ} (Ω : Set (Fin d → ℝ)) : Prop :=
∃ T : Set (Fin d → ℝ), T.Countable ∧
volume (Set.univ \ (⋃ t ∈ T, translateSet Ω t)) = 0 ∧
(∀ {t₁ t₂ : Fin d → ℝ}, t₁ ∈ T → t₂ ∈ T → t₁ ≠ t₂ →
volume (translateSet Ω t₁ ∩ translateSet Ω t₂) = 0)