| import Mathlib.Analysis.SpecialFunctions.Pow.Real |
| import Mathlib.Algebra.Order.Archimedean.Basic |
| import Mathlib.Tactic |
|
|
| namespace OAI |
|
|
| /-! # Scalar dyadic selection for comparable positive displacements |
|
|
| The arithmetic is independent of dimension, angular energy and distance-set |
| estimates. |
| -/ |
|
|
| namespace PlanarFalconer |
| open Set Metric |
|
|
| theorem exists_displacement_depth {d : ℝ} (hd : 0 < d) : |
| ∃ n : ℤ, (2:ℝ)^(-(n:ℝ)-1) < d ∧ d ≤ (2:ℝ)^(-(n:ℝ)) := by |
| obtain ⟨m,hm⟩ := exists_mem_Ioc_zpow hd (by norm_num : (1:ℝ) < 2) |
| refine ⟨-(m+1),?_,?_⟩ |
| · have he : -((-(m+1):ℤ):ℝ)-1 = (m:ℝ) := by push_cast; ring |
| rw [he,Real.rpow_intCast] |
| exact hm.1 |
| · have he : -((-(m+1):ℤ):ℝ) = ((m+1:ℤ):ℝ) := by push_cast; ring |
| rw [he,Real.rpow_intCast] |
| exact hm.2 |
|
|
| theorem comparable_displacement_depth {d₁ d₂ : ℝ} {a h : ℕ} |
| (hupper : max d₁ d₂ ≤ 2*(2:ℝ)^(-(a:ℝ))) |
| (hfar : 16*(2:ℝ)^(-(h:ℝ)) < max d₁ d₂) |
| (hc₁ : max d₁ d₂ ≤ 2*d₁) (hc₂ : max d₁ d₂ ≤ 2*d₂) : |
| ∃ b : ℕ, a+9 ≤ b ∧ b ≤ h+5 ∧ |
| (256*(2:ℝ)^(-(b:ℝ)) < d₁ ∧ d₁ ≤ 1024*(2:ℝ)^(-(b:ℝ))) ∧ |
| (256*(2:ℝ)^(-(b:ℝ)) < d₂ ∧ d₂ ≤ 1024*(2:ℝ)^(-(b:ℝ))) := by |
| have hd : 0 < max d₁ d₂ := (by positivity : 0 < 16*(2:ℝ)^(-(h:ℝ))).trans hfar |
| obtain ⟨n,hnlo,hnhi⟩ := exists_displacement_depth hd |
| have haexp : (2:ℝ)^(-(n:ℝ)-1) < (2:ℝ)^(1-(a:ℝ)) := by |
| have he : (2:ℝ)^(1-(a:ℝ)) = 2*(2:ℝ)^(-(a:ℝ)) := by |
| rw [sub_eq_add_neg,Real.rpow_add (by norm_num : (0:ℝ) < 2),Real.rpow_one] |
| rw [he] |
| exact hnlo.trans_le hupper |
| have haR := (Real.rpow_lt_rpow_left_iff (by norm_num : (1:ℝ) < 2)).mp haexp |
| have hhExp : (2:ℝ)^(4-(h:ℝ)) < (2:ℝ)^(-(n:ℝ)) := by |
| have he : (2:ℝ)^(4-(h:ℝ)) = 16*(2:ℝ)^(-(h:ℝ)) := by |
| rw [sub_eq_add_neg,Real.rpow_add (by norm_num : (0:ℝ) < 2)] |
| norm_num |
| rw [he] |
| exact hfar.trans_le hnhi |
| have hhR := (Real.rpow_lt_rpow_left_iff (by norm_num : (1:ℝ) < 2)).mp hhExp |
| have haZ : (a:ℤ)-2 < n := by exact_mod_cast (show (a:ℝ)-2 < (n:ℝ) by linarith) |
| have hhZ : n < (h:ℤ)-4 := by exact_mod_cast (show (n:ℝ) < (h:ℝ)-4 by linarith) |
| have hb0 : 0 ≤ n+10 := by omega |
| let b := (n+10).toNat |
| have hbZ : (b:ℤ) = n+10 := Int.toNat_of_nonneg hb0 |
| have hbR : (b:ℝ) = (n:ℝ)+10 := by exact_mod_cast hbZ |
| have hpow : (2:ℝ)^(-(n:ℝ)) = 1024*(2:ℝ)^(-(b:ℝ)) := by |
| rw [hbR,show -(n:ℝ) = 10+(-((n:ℝ)+10)) by ring, |
| Real.rpow_add (by norm_num : (0:ℝ) < 2)] |
| norm_num |
| have hhalf : (2:ℝ)^(-(n:ℝ)-1) = 512*(2:ℝ)^(-(b:ℝ)) := by |
| rw [hbR,show -(n:ℝ)-1 = 9+(-((n:ℝ)+10)) by ring, |
| Real.rpow_add (by norm_num : (0:ℝ) < 2)] |
| norm_num |
| rw [hpow] at hnhi |
| rw [hhalf] at hnlo |
| refine ⟨b,?_,?_,?_,?_⟩ |
| · omega |
| · omega |
| · exact ⟨by linarith, (le_max_left _ _).trans hnhi⟩ |
| · exact ⟨by linarith, (le_max_right _ _).trans hnhi⟩ |
|
|
| end PlanarFalconer |
|
|
| end OAI |
|
|
