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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 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 | import OAI.MeasureTheory.Falconer.Estimates.Assembly
import Mathlib.Data.Fin.Basic
import Mathlib.Tactic.FinCases
namespace OAI
/-!
# Entry depth for the all-dimensional assembly
This is the numerical content of `asm:entry`. The dimension enters through
the Lipschitz constant of the two excess profiles. The envelope and
dimension constants are independent.
-/
namespace Falconer
open Set
/-- A common entry depth with a small graph cost and two positive tail
potentials. The large-scale hypothesis is explicit; the manuscript uses
this only for all sufficiently large `N`. -/
theorem exists_assembly_entry (A : Fin 2 → ℕ → ℝ) (N Q : ℕ)
{d χ β ε : ℝ} (hd : 0 < d) (hχ : 0 < χ) (hβ : 0 < β)
(hε : 0 ≤ ε) (hsmall : β + ε < χ / 4)
(hN : 0 < N) (hlarge : 2 * d ≤ β * (N : ℝ))
(hQ : 0 < Q) (hQsmall : 4 * Q ≤ N)
(hzero : ∀ i, A i 0 = 0)
(henvelope : ∀ i n, n ≤ N → χ * (n : ℝ) / N - ε ≤ A i n)
(hlip : ∀ i m n, m ≤ N → n ≤ N →
|A i n - A i m| ≤ d * |(n : ℝ) - m| / N) :
∃ c : ℕ, c < N ∧ β * N / (2 * d) ≤ (c : ℝ) ∧ (c : ℝ) < N / 4 ∧
c + Q ≤ N ∧
(∀ i n, c ≤ n → n ≤ N → β - d / N ≤ A i n) ∧
min (PlanarFalconer.height (A 0) 0 c) (PlanarFalconer.height (A 1) 0 c) ≤ β + ε ∧
(∀ i, 2 * β - 2 * d / N ≤ PlanarFalconer.potential (A i) c N Q) ∧
-A 0 c - A 1 c +
min (PlanarFalconer.height (A 0) 0 c) (PlanarFalconer.height (A 1) 0 c) +
((2 * A 0 c - PlanarFalconer.potential (A 0) c N Q) +
(2 * A 1 c - PlanarFalconer.potential (A 1) c N Q)) / 2 ≤
-β + ε + 2 * d / N := by
have hNr : 0 < (N : ℝ) := by exact_mod_cast hN
let B : ℕ → ℝ := fun n => min (A 0 n) (A 1 n)
obtain ⟨c, hcN, hc, hlast⟩ := PlanarFalconer.exists_last_crossing B N β (by
simp only [B, hzero, min_self]
exact hβ.le)
have hBN : β < B N := by
have h0 := henvelope 0 N le_rfl
have h1 := henvelope 1 N le_rfl
have hdiv : χ * (N : ℝ) / N = χ := by field_simp
rw [hdiv] at h0 h1
exact lt_min (by linarith) (by linarith)
have hstep (n : ℕ) (hn : n < N) : B (n+1) ≤ B n + d / N := by
have hi (i : Fin 2) : A i (n+1) ≤ A i n + d / N := by
have hh := hlip i n (n+1) hn.le (by omega)
have heq : |((n+1 : ℕ) : ℝ) - n| = 1 := by
rw [Nat.cast_add, Nat.cast_one, add_sub_cancel_left, abs_one]
rw [heq, mul_one] at hh
linarith [(abs_le.mp hh).2]
exact PlanarFalconer.min_step (A 0) (A 1) (hi 0) (hi 1)
obtain ⟨hcN', htail⟩ := PlanarFalconer.last_crossing_tail B hcN hc hBN
(div_nonneg hd.le hNr.le) hlast hstep
have hlowall (i : Fin 2) (n : ℕ) (hn : n ≤ N) : -ε ≤ A i n := by
have hh := henvelope i n hn
have hp : 0 ≤ χ * (n : ℝ) / N := by positivity
linarith
have htail' (i : Fin 2) (n : ℕ) (hcn : c ≤ n) (hn : n ≤ N) :
β - d / N ≤ A i n := by
have hh := htail n hcn hn
fin_cases i
· exact hh.trans (min_le_left _ _)
· exact hh.trans (min_le_right _ _)
have hnext := hlast (c+1) (by omega) (by omega)
have hu := hlip 0 0 (c+1) (Nat.zero_le N) (by omega)
rw [hzero, sub_zero, Nat.cast_zero, sub_zero, abs_of_nonneg (show 0 ≤ ((c+1 : ℕ) : ℝ) from Nat.cast_nonneg _)] at hu
have hnext0 : β < A 0 (c+1) := hnext.trans_le (min_le_left _ _)
have hbound : β * (N : ℝ) < d * ((c+1 : ℕ) : ℝ) :=
(lt_div_iff₀ hNr).mp (hnext0.trans_le (le_abs_self _ |>.trans hu))
have hdepthlo : β * N / (2 * d) ≤ (c : ℝ) := by
apply (div_le_iff₀ (by positivity : 0 < 2 * d)).mpr
push_cast at hbound
nlinarith
have henvmin : χ * (c : ℝ) / N - ε ≤ B c :=
le_min (henvelope 0 c hcN) (henvelope 1 c hcN)
have hdepthhi : (c : ℝ) < (N : ℝ) / 4 := by
have hdiv : χ * (c : ℝ) / N < χ / 4 := by linarith
have hm := (div_lt_iff₀ hNr).mp hdiv
nlinarith
have hwidth : c + Q ≤ N := by
have hQreal : (4 : ℝ) * Q ≤ N := by exact_mod_cast hQsmall
have hcQ : (c : ℝ) + Q ≤ N := by linarith
exact_mod_cast hcQ
have hcost := PlanarFalconer.min_height_le (A 0) (A 1) c hc
(fun n hn => hlowall 0 n (hn.trans hcN))
(fun n hn => hlowall 1 n (hn.trans hcN))
have hpot (i : Fin 2) : 2 * β - 2 * d / N ≤ PlanarFalconer.potential (A i) c N Q := by
have hh := PlanarFalconer.potential_lower (A i) hQ hwidth (htail' i)
convert hh using 1; ring
refine ⟨c, hcN', hdepthlo, hdepthhi, hwidth, htail', hcost, hpot, ?_⟩
have hg := PlanarFalconer.outer_profile_gain (A 0) (A 1) hQ hwidth hc
(fun n hn => hlowall 0 n (hn.trans hcN))
(fun n hn => hlowall 1 n (hn.trans hcN)) htail
convert hg using 1; ring
end Falconer
end OAI
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