| import OAI.Analysis.IrrationalRotation.Rotation.FiniteSupportWeights |
| import OAI.Analysis.IrrationalRotation.Rotation.CircleRepresentative |
| import Mathlib.Dynamics.Ergodic.AddCircle |
| import Mathlib.MeasureTheory.Constructions.Polish.Basic |
|
|
| namespace OAI |
|
|
| open MeasureTheory Set |
| open scoped ENNReal |
|
|
| namespace Invariant233.CircleWeightSupport |
|
|
| theorem measurePreserving_characterRepresentative (q : ℤ) (hq : q ≠ 0) : |
| MeasurePreserving |
| (fun x : UnitAddCircle => ((AddCircle.equivIco (1 : ℝ) 0 (q • x)) : ℝ)) |
| volume (volume.restrict (Ico (0 : ℝ) 1)) := by |
| exact CocycleSupport.measurePreserving_circleRepresentative.comp |
| (Measure.measurePreserving_zsmul volume hq) |
|
|
| theorem ae_all_int_finite_floor_support {ι : Type*} [Countable ι] |
| (q : ι → ℤ) (hq : ∀ i, q i ≠ 0) |
| (α : ι → ℝ) (hα0 : ∀ i, 0 ≤ α i) (hα : Summable α) : |
| ∀ᵐ x ∂(volume : Measure UnitAddCircle), ∀ k : ℤ, |
| (Function.support (fun i => |
| Int.floor (((AddCircle.equivIco (1 : ℝ) 0 (q i • x)) : ℝ) + |
| (k : ℝ) * α i))).Finite := by |
| exact FiniteSupportWeights.ae_all_finite_floor_support volume |
| (fun i x => ((AddCircle.equivIco (1 : ℝ) 0 (q i • x)) : ℝ)) |
| (fun i => measurePreserving_characterRepresentative (q i) (hq i)) α hα0 hα |
|
|
| theorem exists_invariant_conull_floor_support {ι : Type*} [Countable ι] |
| (θ : ℝ) (q : ι → ℤ) (hq : ∀ i, q i ≠ 0) |
| (α : ι → ℝ) (hα0 : ∀ i, 0 ≤ α i) (hα : Summable α) : |
| ∃ s : Set UnitAddCircle, MeasurableSet s ∧ (∀ᵐ x ∂volume, x ∈ s) ∧ |
| (∀ (n : ℤ) x, x + n • (θ : UnitAddCircle) ∈ s ↔ x ∈ s) ∧ |
| ∀ x ∈ s, ∀ k : ℤ, |
| (Function.support (fun i => |
| Int.floor (((AddCircle.equivIco (1 : ℝ) 0 (q i • x)) : ℝ) + |
| (k : ℝ) * α i))).Finite := by |
| apply FiniteSupportWeights.exists_invariant_conull_floor_support |
| (volume : Measure UnitAddCircle) (fun n x => x + n • (θ : UnitAddCircle)) |
| (fun n => measurePreserving_add_right volume (n • (θ : UnitAddCircle))) |
| (by intro x; simp) |
| (by intro m n x; simp [add_zsmul, add_comm, add_left_comm]) |
| (fun i x => ((AddCircle.equivIco (1 : ℝ) 0 (q i • x)) : ℝ)) |
| (fun i => measurePreserving_characterRepresentative (q i) (hq i)) α hα0 hα |
|
|
| theorem measurable_exp_weight {ι : Type*} [Countable ι] |
| (q : ι → ℤ) (hq : ∀ i, q i ≠ 0) (α c : ι → ℝ) : |
| Measurable (fun x : UnitAddCircle => Real.exp (-(∑' i, |
| c i * (Int.floor (((AddCircle.equivIco (1 : ℝ) 0 (q i • x)) : ℝ) + α i) : ℝ)))) := by |
| apply Measurable.exp |
| apply Measurable.neg |
| apply Measurable.tsum |
| intro i |
| apply Measurable.mul measurable_const |
| exact (measurable_of_countable (fun z : ℤ => (z : ℝ))).comp |
| (((measurePreserving_characterRepresentative (q i) (hq i)).measurable.add_const _).floor) |
| |
| theorem ae_summable_weighted_floors {ι : Type*} [Countable ι] |
| (q : ι → ℤ) (hq : ∀ i, q i ≠ 0) |
| (α : ι → ℝ) (hα0 : ∀ i, 0 ≤ α i) (hα : Summable α) (c : ι → ℝ) : |
| ∀ᵐ x ∂(volume : Measure UnitAddCircle), |
| Summable (fun i => c i * |
| (Int.floor (((AddCircle.equivIco (1 : ℝ) 0 (q i • x)) : ℝ) + α i) : ℝ)) := by |
| filter_upwards [ae_all_int_finite_floor_support q hq α hα0 hα] with x hx |
| apply FiniteSupportWeights.summable_mul_cost |
| have hf : (Function.support (fun i => |
| Int.floor (((AddCircle.equivIco (1 : ℝ) 0 (q i • x)) : ℝ) + α i))).Finite := by |
| simpa only [Int.cast_one, one_mul] using hx 1 |
| apply hf.subset |
| intro i hi |
| simp only [Function.mem_support] at hi ⊢ |
| intro h |
| exact hi (by simp [h]) |
| |
| theorem exp_weight_mem_Ioc {ι : Type*} (q : ι → ℤ) (α c : ι → ℝ) |
| (hα0 : ∀ i, 0 ≤ α i) (hc : ∀ i, 0 ≤ c i) (x : UnitAddCircle) : |
| Real.exp (-(∑' i, |
| c i * (Int.floor (((AddCircle.equivIco (1 : ℝ) 0 (q i • x)) : ℝ) + α i) : ℝ))) |
| ∈ Set.Ioc (0 : ℝ) 1 := by |
| apply FiniteSupportWeights.exp_neg_tsum_mem_Ioc c _ hc |
| intro i |
| apply Int.cast_nonneg |
| apply Int.floor_nonneg.mpr |
| exact add_nonneg (AddCircle.equivIco (1 : ℝ) 0 (q i • x)).property.1 (hα0 i) |
|
|
| end Invariant233.CircleWeightSupport |
|
|
| end OAI |
|
|
