| import Mathlib.Analysis.Fourier.AddCircleMulti |
| import Mathlib.MeasureTheory.Measure.ProbabilityMeasure |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace Invariant233.CharacterLaws |
|
|
| open MeasureTheory |
|
|
| def circleHaar : ProbabilityMeasure (AddCircle (1 : ℝ)) := |
| ⟨volume, ⟨by simp [AddCircle.measure_univ]⟩⟩ |
|
|
| def torusHaar (m : ℕ) : ProbabilityMeasure (UnitAddTorus (Fin m)) := by |
| letI : IsProbabilityMeasure (volume : Measure (AddCircle (1 : ℝ))) := |
| circleHaar.property |
| exact ⟨volume, inferInstance⟩ |
|
|
| def characterMap {m : ℕ} (q : Fin m → ℕ) : |
| C(AddCircle (1 : ℝ), UnitAddTorus (Fin m)) := |
| ⟨fun x i => (q i : ℤ) • x, by fun_prop⟩ |
|
|
| def characterLaw {m : ℕ} (q : Fin m → ℕ) : ProbabilityMeasure (UnitAddTorus (Fin m)) := |
| circleHaar.map (characterMap q) |
|
|
| def characterPhaseLaw {Z : Type*} [TopologicalSpace Z] [MeasurableSpace Z] |
| [BorelSpace Z] {m : ℕ} (q : Fin m → ℕ) (a : Z) : |
| ProbabilityMeasure (Z × UnitAddTorus (Fin m)) := |
| circleHaar.map (fun x => (a, characterMap q x)) |
|
|
| def haarPhaseLaw {Z : Type*} [TopologicalSpace Z] [MeasurableSpace Z] |
| [BorelSpace Z] (m : ℕ) (a : Z) : ProbabilityMeasure (Z × UnitAddTorus (Fin m)) := |
| (torusHaar m).map (fun y => (a, y)) |
|
|
| end Invariant233.CharacterLaws |
|
|
| end |
|
|
| end OAI |
|
|
