| import OAI.Analysis.IrrationalRotation.Model.MixedRadixCircle |
| import OAI.Analysis.IrrationalRotation.Sampling.MixedRadix |
| import OAI.Analysis.IrrationalRotation.Analysis.DensityTransport |
|
|
| namespace OAI |
|
|
| open MeasureTheory |
| open scoped ENNReal |
|
|
| namespace Invariant233.MixedRadix |
|
|
| theorem measurePreserving_unitCircle : |
| MeasurePreserving (fun x : ℝ => (x : AddCircle (1 : ℝ))) |
| (volume.restrict (Set.Ico 0 1)) volume := by |
| rw [restrict_Ico_eq_restrict_Ioc] |
| simpa using AddCircle.measurePreserving_mk (1 : ℝ) 0 |
|
|
| theorem measurable_circleValue {r : ℕ → ℕ} (hr : ∀ n, 2 ≤ r n) : |
| Measurable (@circleValue r) := |
| AddCircle.measurable_mk'.comp (measurable_ofDigits hr) |
|
|
| theorem measurable_circleCanonical (r : ℕ → ℕ) (hr : ∀ n, 0 < r n) : |
| Measurable (circleCanonical r hr) := |
| (measurable_canonicalDigits r hr).comp |
| (measurable_subtype_coe.comp (AddCircle.measurableEquivIco (1 : ℝ) 0).measurable) |
|
|
| theorem circleValue_circleCanonical {r : ℕ → ℕ} (hr : ∀ n, 2 ≤ r n) |
| (hp : ∀ n, 0 < r n) (x : AddCircle (1 : ℝ)) : |
| circleValue (circleCanonical r hp x) = x := by |
| have hx : (AddCircle.equivIco (1 : ℝ) 0 x : ℝ) ∈ Set.Ico 0 1 := by |
| simpa using (AddCircle.equivIco (1 : ℝ) 0 x).property |
| unfold circleValue circleCanonical |
| rw [ofDigits_canonicalDigits hr hx] |
| exact AddCircle.coe_equivIco |
|
|
| theorem circleCanonical_circleValue {r : ℕ → ℕ} (hp : ∀ n, 0 < r n) |
| (d : Digits r) (hnt : Nonterminal d) : |
| circleCanonical r hp (circleValue d) = d := by |
| have hx : ofDigits d ∈ Set.Ico (0 : ℝ) (0 + 1) := by |
| simpa using ofDigits_mem_Ico hp d hnt |
| unfold circleCanonical circleValue |
| rw [AddCircle.equivIco_coe_eq hx] |
| exact canonicalDigits_ofDigits hp d hnt |
|
|
| theorem circleValue_measurePreserving {r : ℕ → ℕ} (hr : ∀ n, 2 ≤ r n) |
| (μ : Measure (Digits r)) |
| (hlaw : μ.map ofDigits = volume.restrict (Set.Ico 0 1)) : |
| MeasurePreserving (@circleValue r) μ volume := by |
| exact measurePreserving_unitCircle.comp ⟨measurable_ofDigits hr, hlaw⟩ |
|
|
| theorem circle_density_transport {r : ℕ → ℕ} (hr : ∀ n, 2 ≤ r n) |
| (hp : ∀ n, 0 < r n) (μ : Measure (Digits r)) |
| (hlaw : μ.map ofDigits = volume.restrict (Set.Ico 0 1)) |
| (hnt : ∀ᵐ d ∂μ, Nonterminal d) {h : Digits r → ℝ} (hh : Measurable h) : |
| (μ.withDensity (fun d => ENNReal.ofReal (h d))).map circleValue = |
| volume.withDensity (fun x : AddCircle (1 : ℝ) => |
| ENNReal.ofReal (h (circleCanonical r hp x))) ∧ |
| eLpNorm (h ∘ circleCanonical r hp) 2 volume = eLpNorm h 2 μ := by |
| apply DensityTransport.real_density_transport (measurable_circleValue hr) |
| (measurable_circleCanonical r hp) (circleValue_measurePreserving hr μ hlaw).map_eq |
| · filter_upwards [hnt] with d hd |
| exact circleCanonical_circleValue hp d hd |
| · exact hh |
|
|
| end Invariant233.MixedRadix |
|
|
| end OAI |
|
|
