| import Mathlib |
| import OAI.Probability.CriticalPercolation.Growth.RealSparseDoubling |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace CriticalPercolation.HarmonicAnalysis |
|
|
| lemma exists_small_relative_dissipation (E N : ℕ → ℝ) |
| (hpos : ∀ n, 0<E n) (hm : Antitone E) |
| (hdis : ∀ n, N n≤E n-E (n+1)) |
| (d : ℕ) (C : ℝ) (hC : 0≤C) |
| (hg : ∀ R, ∃ n≥R, (E n)⁻¹≤C*(n:ℝ)^d) |
| (ε : ℝ) (hε : 0<ε) : ∃ n, N n≤ε*E n := by |
| by_contra! hbad |
| have hmono : Monotone (fun n => (E n)⁻¹) := by |
| intro i j hij |
| exact (inv_le_inv₀ (hpos i) (hpos j)).mpr (hm hij) |
| obtain ⟨R,hR⟩ := exists_nat_gt ((2:ℝ)^(d+1)/ε) |
| obtain ⟨r,hr,hgood⟩ := sparse_real_doubling (fun n => (E n)⁻¹) hmono |
| 0 (fun n _ => inv_pos.mpr (hpos n)) C hC d 2 (by omega) hg R |
| have hratio : E r≤(2:ℝ)^(d+1)*E (2*r) := by |
| have hh : 1/E (2*r)≤(2:ℝ)^(d+1)/E r := by |
| simpa only [Nat.cast_ofNat,one_div,div_eq_mul_inv,one_mul] using hgood |
| have hh' := (div_le_div_iff₀ (hpos (2*r)) (hpos r)).mp hh |
| simpa only [one_mul] using hh' |
| have hεr : (2:ℝ)^(d+1)<(r:ℝ)*ε := by |
| have hh : (R:ℝ)≤r := by exact_mod_cast (le_max_left R 0).trans hr |
| have hh' := (div_lt_iff₀ hε).mp hR |
| nlinarith |
| have ht : ∀ i≤r, (i:ℝ)*ε*E (2*r)≤E r-E (r+i) := by |
| intro i hi |
| induction i with |
| | zero => simp |
| | succ i ih => |
| have H := ih (by omega) |
| have HD := (hbad (r+i)).le.trans (hdis (r+i)) |
| have HM := mul_le_mul_of_nonneg_left (hm (by omega : r+i≤2*r)) hε.le |
| push_cast |
| have he : r+(i+1)=r+i+1 := by omega |
| rw [he] |
| nlinarith |
| have hh := ht r le_rfl |
| rw [←two_mul] at hh |
| have hh' := mul_lt_mul_of_pos_right hεr (hpos (2*r)) |
| nlinarith [hpos (2*r)] |
|
|
| end CriticalPercolation.HarmonicAnalysis |
| end |
| end |
|
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| end OAI |
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