| import OAI.Analysis.IrrationalRotation.Sampling.BlockPhaseConstruction |
| import OAI.Analysis.IrrationalRotation.Product.ProductBlockCost |
|
|
| namespace OAI |
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|
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped BigOperators Topology |
|
|
| namespace Invariant233.ProductModel |
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| open ProductPhases |
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|
| theorem exists_admissible_phase_block_weight_and |
| (m : ℕ → ℕ) (hgrowth : ∀ j, 2 ^ j ≤ m j) |
| (K G C : ℕ → ℝ) (hK : ∀ j, 1 ≤ K j) (hG : ∀ j, 0 < G j) |
| (hC : ∀ j, (K j) ^ 2 ≤ C j) |
| (hscale : ∀ j, 2 < (m j : ℝ) * K j) |
| (hsum : Summable (fun j => 1 / K j)) |
| (e : ℕ ≃ Σ j, Fin (m j)) |
| (he : ∀ a : Σ j, Fin (m j), |
| e.symm a = (∑ r ∈ Finset.range a.1, m r) + a.2.val) |
| {P : (ℕ → ℝ) → Prop} |
| (hP : ∀ᵐ θ ∂Measure.infinitePi (fun i => ProbabilityTheory.cond volume |
| (Icc (1 / ((m (e i).1 : ℝ) * K (e i).1)) |
| (2 / ((m (e i).1 : ℝ) * K (e i).1)))), P θ) : |
| ∃ θ : ℕ → ℝ, |
| (∀ j, ∀ i ∈ consecutiveBlock m j, |
| θ i ∈ Icc (1 / ((m j : ℝ) * K j)) (2 / ((m j : ℝ) * K j))) ∧ |
| (∑' i, θ i) ≤ 2 * (∑' j, 1 / K j) ∧ |
| AdmissibleAnglesAndCosts θ (fun i => C (e i).1) ∧ |
| HasNegInfiniteLogIntegral baseMeasure (weight θ (fun i => C (e i).1)) ∧ |
| (∀ᵐ x ∂baseMeasure, |
| 0 < weight θ (fun i => C (e i).1) x ∧ weight θ (fun i => C (e i).1) x ≤ 1) ∧ |
| (∀ j, smallPhaseCondition (consecutiveBlock m j) (K j) (G j) θ) ∧ |
| (∀ᶠ j in atTop, |
| largePhaseCondition (consecutiveBlock m j) ((m j : ℝ) * K j) θ) ∧ P θ := by |
| have hm : ∀ j, 0 < m j := fun j => |
| lt_of_lt_of_le (by positivity : 0 < 2 ^ j) (hgrowth j) |
| have hKpos (j : ℕ) : 0 < K j := zero_lt_one.trans_le (hK j) |
| have hCpos (j : ℕ) : 0 < C j := (sq_pos_of_pos (hKpos j)).trans_le (hC j) |
| obtain ⟨θ, hθ, hpos, hsθ, htθ, hind, hsmall, hlarge, hp⟩ := |
| exists_consecutive_phase_angles_and m hgrowth K G hKpos hG hsum e he hP |
| have hbounds (i : ℕ) : θ i ∈ |
| Icc (1 / ((m (e i).1 : ℝ) * K (e i).1)) |
| (2 / ((m (e i).1 : ℝ) * K (e i).1)) := |
| hθ (e i).1 i (mem_consecutiveBlock_enumeration m e he i) |
| have hunit (i : ℕ) : θ i < 1 := |
| (hbounds i).2.trans_lt ((div_lt_one |
| (mul_pos (Nat.cast_pos.mpr (hm _)) (hKpos _))).mpr (hscale _)) |
| refine ⟨θ, hθ, htθ, ?_, ?_, ?_, hsmall, hlarge, hp⟩ |
| · exact ⟨fun i => ⟨hpos i, hunit i, hCpos _⟩, hsθ, hind⟩ |
| · exact hasNegInfiniteLogIntegral_of_block_bounds m hm K C hK hC e θ |
| (fun i => ⟨(hpos i).le, hunit i⟩) (fun i => (hbounds i).1) |
| · filter_upwards [ae_weight_pos θ (fun i => C (e i).1) |
| (fun i => (hpos i).le) (fun i => (hCpos _).le) hsθ] with x hx |
| exact ⟨hx, (weight_mem_Icc θ _ x).2⟩ |
|
|
| theorem exists_admissible_phase_block_weight |
| (m : ℕ → ℕ) (hgrowth : ∀ j, 2 ^ j ≤ m j) |
| (K G C : ℕ → ℝ) (hK : ∀ j, 1 ≤ K j) (hG : ∀ j, 0 < G j) |
| (hC : ∀ j, (K j) ^ 2 ≤ C j) |
| (hscale : ∀ j, 2 < (m j : ℝ) * K j) |
| (hsum : Summable (fun j => 1 / K j)) : |
| ∃ (e : ℕ ≃ Σ j, Fin (m j)) (θ : ℕ → ℝ), |
| (∀ a : Σ j, Fin (m j), |
| e.symm a = (∑ r ∈ Finset.range a.1, m r) + a.2.val) ∧ |
| (∀ j, ∀ i ∈ consecutiveBlock m j, |
| θ i ∈ Icc (1 / ((m j : ℝ) * K j)) (2 / ((m j : ℝ) * K j))) ∧ |
| (∑' i, θ i) ≤ 2 * (∑' j, 1 / K j) ∧ |
| AdmissibleAnglesAndCosts θ (fun i => C (e i).1) ∧ |
| HasNegInfiniteLogIntegral baseMeasure (weight θ (fun i => C (e i).1)) ∧ |
| (∀ᵐ x ∂baseMeasure, |
| 0 < weight θ (fun i => C (e i).1) x ∧ weight θ (fun i => C (e i).1) x ≤ 1) ∧ |
| (∀ j, smallPhaseCondition (consecutiveBlock m j) (K j) (G j) θ) ∧ |
| (∀ᶠ j in atTop, |
| largePhaseCondition (consecutiveBlock m j) ((m j : ℝ) * K j) θ) := by |
| have hm : ∀ j, 0 < m j := fun j => |
| lt_of_lt_of_le (by positivity : 0 < 2 ^ j) (hgrowth j) |
| obtain ⟨e, he⟩ := InvariantProjection.IndependentAngles.exists_consecutiveBlocks_equiv m hm |
| obtain ⟨θ, hθ, htθ, hadm, hlog, hweight, hsmall, hlarge, _⟩ := |
| exists_admissible_phase_block_weight_and m hgrowth K G C hK hG hC hscale hsum e he |
| (P := fun _ => True) (Filter.Eventually.of_forall fun _ => True.intro) |
| exact ⟨e, θ, he, hθ, htθ, hadm, hlog, hweight, hsmall, hlarge⟩ |
|
|
| end Invariant233.ProductModel |
|
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| end |
|
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| end OAI |
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|
