| import OAI.MeasureTheory.Falconer.Estimates.RestorationSeries |
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| namespace OAI |
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| /-! # Converting the original-pair deletion bound to its summable majorant -/ |
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| namespace Falconer |
| open scoped ENNReal |
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| theorem dyadic_leaf_error (N : ℕ) : |
| ((2:ℝ≥0∞)^(2*N))⁻¹ = ENNReal.ofReal (((2:ℝ)^N)^(-2:ℝ)) := by |
| have hfin : ((2:ℝ≥0∞)^(2*N))⁻¹ ≠ ∞ := by simp |
| rw [← ENNReal.ofReal_toReal hfin] |
| congr 1 |
| simp only [ENNReal.toReal_inv,ENNReal.toReal_pow,ENNReal.toReal_ofNat] |
| rw [Real.rpow_neg (by positivity),Real.rpow_two] |
| congr 1 |
| rw [← pow_mul,Nat.mul_comm N 2] |
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| theorem class_discard_numeric (N k : ℕ) (lam : ℝ) : |
| ((2:ℝ≥0∞)^(2*N))⁻¹ + (N^k:ℕ)*ENNReal.ofReal (((2:ℝ)^N)^(-lam)) ≤ |
| ENNReal.ofReal (((2:ℝ)^N)^(-2:ℝ) + ((N:ℝ)+1)^k*((2:ℝ)^N)^(-lam)) := by |
| have hp : ((N^k:ℕ):ℝ≥0∞) ≤ ENNReal.ofReal (((N:ℝ)+1)^k) := by |
| rw [← ENNReal.ofReal_natCast] |
| apply ENNReal.ofReal_le_ofReal |
| simpa only [Nat.cast_pow] using pow_le_pow_left₀ (Nat.cast_nonneg N) |
| (show (N:ℝ) ≤ N+1 by linarith) k |
| rw [dyadic_leaf_error] |
| apply (add_le_add le_rfl (mul_le_mul' hp le_rfl)).trans_eq |
| rw [← ENNReal.ofReal_mul (by positivity),← ENNReal.ofReal_add (by positivity) (by positivity)] |
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| theorem restoration_numeric_bound (N k₁ k₂ : ℕ) {Cw Ccap c lam σ : ℝ} |
| (hw : 0 ≤ Cw) (hcap : 0 ≤ Ccap) {D X₁ X₂ V : ℝ≥0∞} |
| (hD : D ≤ ENNReal.ofReal (Ccap*((2:ℝ)^N)^(-c))) |
| (h₁ : X₁ ≤ ((2:ℝ≥0∞)^(2*N))⁻¹ + (N^k₁:ℕ)*ENNReal.ofReal (((2:ℝ)^N)^(-lam))) |
| (h₂ : X₂ ≤ ((2:ℝ≥0∞)^(2*N))⁻¹ + (N^k₂:ℕ)*ENNReal.ofReal (((2:ℝ)^N)^(-lam))) |
| (hV : V ≤ ENNReal.ofReal Cw * (D + (X₁ + X₂ + |
| (4*(N+1)^2:ℕ)*ENNReal.ofReal (((2:ℝ)^N)^(-σ))))) : |
| V ≤ ENNReal.ofReal (restorationError Cw Ccap c lam σ k₁ k₂ N) := by |
| apply hV.trans |
| have ht : (4*(N+1)^2:ℕ)*ENNReal.ofReal (((2:ℝ)^N)^(-σ)) = |
| ENNReal.ofReal (4*((N:ℝ)+1)^2*((2:ℝ)^N)^(-σ)) := by |
| rw [← ENNReal.ofReal_natCast,← ENNReal.ofReal_mul (by positivity)] |
| simp only [Nat.cast_mul,Nat.cast_ofNat,Nat.cast_pow,Nat.cast_add,Nat.cast_one] |
| rw [ht,restorationError,ENNReal.ofReal_mul hw] |
| apply mul_le_mul' le_rfl |
| apply (add_le_add hD (add_le_add (add_le_add (h₁.trans (class_discard_numeric N k₁ lam)) |
| (h₂.trans (class_discard_numeric N k₂ lam))) le_rfl)).trans_eq |
| rw [← ENNReal.ofReal_add (by positivity) (by positivity), |
| ← ENNReal.ofReal_add (by positivity) (by positivity), |
| ← ENNReal.ofReal_add (by positivity) (by positivity)] |
| congr 1 |
| ring |
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| end Falconer |
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| end OAI |
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