| 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 |
|
|
| / |
| 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 |
|
|
