| import OAI.MeasureTheory.Falconer.Geometry.CommonProjectionConfiguration |
|
|
| namespace OAI |
|
|
| /-! A single weak constant controls every loss from selecting the common |
| projection bases and restricting to their intersection. -/ |
| namespace Falconer |
|
|
| theorem common_projection_losses {W : ℝ} (hW : 1 ≤ W) : |
| let ρ := W⁻¹ |
| let Z := 64*W^6 |
| 0 < ρ ∧ ρ ≤ 1 ∧ 1 ≤ Z ∧ W ≤ Z ∧ W*(8/ρ^4) ≤ Z ∧ |
| 64/ρ^6 = Z ∧ 0 < ρ^4/8 ∧ (ρ^4/8)⁻¹ ≤ Z := by |
| have hWpos : 0 < W := lt_of_lt_of_le zero_lt_one hW |
| have hρ : 0 < W⁻¹ := inv_pos.mpr hWpos |
| have hρ1 : W⁻¹ ≤ 1 := (inv_le_one₀ hWpos).2 hW |
| have h6 : 1 ≤ W^6 := one_le_pow₀ hW |
| have hW6 : W ≤ W^6 := by simpa only [pow_one] using pow_le_pow_right₀ hW (show 1 ≤ (6:ℕ) by omega) |
| have hW5 : W^5 ≤ W^6 := pow_le_pow_right₀ hW (by omega) |
| have hW4 : W^4 ≤ W^6 := pow_le_pow_right₀ hW (by omega) |
| have he4 : 8/(W⁻¹)^4 = 8*W^4 := by field_simp |
| have he6 : 64/(W⁻¹)^6 = 64*W^6 := by field_simp |
| have hei : ((W⁻¹)^4/8)⁻¹ = 8*W^4 := by field_simp |
| refine ⟨hρ,hρ1,by nlinarith,by nlinarith,?_,he6,by positivity,?_⟩ |
| · rw [he4] |
| have he : W*(8*W^4) = 8*W^5 := by ring |
| rw [he] |
| nlinarith |
| · rw [hei] |
| nlinarith |
|
|
| theorem common_projection_direction_count {ρ : ℝ} (hρ : 0 < ρ) |
| (I J : Finset ℝ) (hcount : (ρ^6/64)*(I.card:ℝ) ≤ (J.card:ℝ)) : |
| (I.card:ℝ) ≤ (64/ρ^6)*(J.card:ℝ) := by |
| have hh := mul_le_mul_of_nonneg_left hcount (show 0 ≤ 64/ρ^6 by positivity) |
| have he : (64/ρ^6)*((ρ^6/64)*(I.card:ℝ)) = (I.card:ℝ) := by field_simp |
| rwa [he] at hh |
|
|
| end Falconer |
|
|
| end OAI |
|
|
