| import OAI.MeasureTheory.Falconer.Recursion.ProfileRoundContact |
|
|
| namespace OAI |
|
|
| /-! |
|
|
| namespace Falconer |
| open Filter |
| open scoped Topology |
|
|
| theorem contact_depth_margins {A B : ℝ} (hA : 0 < A) (hAB : A < B) |
| (hB : B < 1) (hBA : B < 2*A) : |
| ∃ β γ : ℝ, 0 < β ∧ 0 < γ ∧ ∃ N₀ : ℕ, ∀ N ≥ N₀, |
| let a := Nat.floor (A*(N:ℝ)) |
| let b := Nat.floor (B*(N:ℝ)) |
| a ≤ b ∧ b ≤ N ∧ β*(N:ℝ) ≤ (b-a:ℕ) ∧ γ*(N:ℝ) ≤ 2*(a:ℝ)-(b:ℝ) := by |
| let β := (B-A)/2 |
| let γ := (2*A-B)/2 |
| have hβ : 0 < β := by dsimp [β]; linarith |
| have hγ : 0 < γ := by dsimp [γ]; linarith |
| have hAlim := floor_depth_ratio_tendsto hA.le |
| have hBlim := floor_depth_ratio_tendsto (hA.trans hAB).le |
| have hgap := (hBlim.sub hAlim).eventually (eventually_gt_nhds (show β < B-A by dsimp [β]; linarith)) |
| have hlinear := ((tendsto_const_nhds.mul hAlim).sub hBlim).eventually |
| (eventually_gt_nhds (show γ < 2*A-B by dsimp [γ]; linarith)) |
| obtain ⟨N₀,hN₀⟩ := eventually_atTop.mp (hgap.and (hlinear.and (eventually_ge_atTop 1))) |
| refine ⟨β,γ,hβ,hγ,N₀,?_⟩ |
| intro N hN |
| obtain ⟨hgap,hlinear,hNpos⟩ := hN₀ N hN |
| have hNp : 0 < (N:ℝ) := by exact_mod_cast (show 0 < N by omega) |
| let a := Nat.floor (A*(N:ℝ)) |
| let b := Nat.floor (B*(N:ℝ)) |
| have hab : a ≤ b := Nat.floor_mono (mul_le_mul_of_nonneg_right hAB.le hNp.le) |
| have hbN : b ≤ N := by |
| apply Nat.floor_le_of_le |
| exact (mul_le_mul_of_nonneg_right hB.le hNp.le).trans_eq (one_mul _) |
| refine ⟨hab,hbN,?_,?_⟩ |
| · rw [Nat.cast_sub hab] |
| apply (le_div_iff₀ hNp).mp |
| simpa only [sub_div] using hgap.le |
| · apply (le_div_iff₀ hNp).mp |
| simpa only [sub_div,mul_div_assoc] using hlinear.le |
|
|
| end Falconer |
|
|
| end OAI |
|
|
