| import OAI.MeasureTheory.Falconer.Slicing.GaussianLossAbsorption | |
| namespace OAI | |
| /-! # The original tube count after codimension-h slicing -/ | |
| namespace Falconer | |
| theorem gaussian_active_cover_scale {δ ε κ q Nc D : ℝ} (h : ℕ) | |
| (hδ : 0 < δ) (hδ1 : δ ≤ 1) (hεκ : ε ≤ κ) (hD : 0 ≤ D) | |
| (hNc : Nc ≤ δ^(-q-ε)) : | |
| (δ^(-κ))^11*(Nc*(2*(D*δ))^h) ≤ | |
| (2*D)^h*(δ^(-κ))^12*δ^(-(q-(h:ℝ))) := by | |
| have hε : δ^(-ε) ≤ δ^(-κ) := | |
| Real.rpow_le_rpow_of_exponent_ge hδ hδ1 (neg_le_neg hεκ) | |
| have hpower : δ^(-q-ε)*δ^h = δ^(-ε)*δ^(-(q-(h:ℝ))) := by | |
| rw [← Real.rpow_natCast,← Real.rpow_add hδ,← Real.rpow_add hδ] | |
| congr 1 | |
| ring | |
| calc | |
| _ ≤ (δ^(-κ))^11*(δ^(-q-ε)*(2*(D*δ))^h) := | |
| mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_right hNc (by positivity)) (by positivity) | |
| _ = (2*D)^h*(δ^(-κ))^11*(δ^(-ε)*δ^(-(q-(h:ℝ)))) := by | |
| rw [show 2*(D*δ) = (2*D)*δ by ring,mul_pow] | |
| calc | |
| _ = (2*D)^h*(δ^(-κ))^11*(δ^(-q-ε)*δ^h) := by ring | |
| _ = _ := by rw [hpower] | |
| _ ≤ (2*D)^h*(δ^(-κ))^11*(δ^(-κ)*δ^(-(q-(h:ℝ)))) := | |
| mul_le_mul_of_nonneg_left | |
| (mul_le_mul_of_nonneg_right hε (Real.rpow_nonneg hδ.le _)) (by positivity) | |
| _ = _ := by ring | |
| end Falconer | |
| end OAI | |