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d4317d7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 | import OAI.MeasureTheory.Falconer.Recursion.ProfileRoundContact
namespace OAI
/-! # Fixed positive margins at the rounded contact depths -/
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
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