| import Mathlib |
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| namespace OAI |
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| section |
| noncomputable section |
| open scoped Classical |
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| namespace CriticalPercolation.HarmonicAnalysis |
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| lemma sparse_real_doubling (f : ℕ → ℝ) (hf : Monotone f) |
| (R₀ : ℕ) (hpos : ∀ n≥R₀, 0<f n) (C : ℝ) (hC : 0≤C) |
| (d b : ℕ) (hb : 2≤b) (hpoly : ∀ R, ∃ n≥R, f n≤C*(n:ℝ)^d) : |
| ∀ R : ℕ, ∃ n≥max R R₀, f (b*n)≤(b:ℝ)^(d+1)*f n := by |
| intro R |
| by_contra! hbad |
| let a := max (max R R₀) 1 |
| have ha : 0<a := by dsimp [a]; omega |
| have haR : max R R₀≤a := le_max_left _ _ |
| have hfapos : 0<f a := hpos a ((le_max_right R R₀).trans haR) |
| have hgrow : ∀ k : ℕ, ((b:ℝ)^(d+1))^k*f a≤f (b^k*a) := by |
| intro k |
| induction k with |
| | zero => simp |
| | succ k ih => |
| have hr : max R R₀≤b^k*a := haR.trans (by |
| have hh : 1≤b^k := Nat.one_le_pow k b (by omega) |
| nlinarith) |
| have hh := (hbad (b^k*a) hr).le |
| calc |
| _ = (b:ℝ)^(d+1)*(((b:ℝ)^(d+1))^k*f a) := by rw [pow_succ]; ring |
| _ ≤ (b:ℝ)^(d+1)*f (b^k*a) := mul_le_mul_of_nonneg_left ih (by positivity) |
| _ ≤ f (b*(b^k*a)) := hh |
| _ = _ := by congr 1; rw [pow_succ]; ring |
| obtain ⟨N,hN⟩ := exists_nat_gt (C*((b*a:ℕ):ℝ)^d/f a) |
| have hNF : C*((b*a:ℕ):ℝ)^d<(N:ℝ)*f a := (div_lt_iff₀ hfapos).mp hN |
| obtain ⟨n,hn,hnpoly⟩ := hpoly (b*a*(N+1)) |
| have hnA : b*(N+1)≤n/a := (Nat.le_div_iff_mul_le ha).mpr (by nlinarith) |
| have hnAp : n/a≠0 := by nlinarith |
| let k := Nat.log b (n/a) |
| let P := b^k |
| have hP : P≤n/a := Nat.pow_log_le_self b hnAp |
| have hP' : n/a<b*P := by |
| have hh := Nat.lt_pow_of_log_lt (b := b) (x := k+1) (y := n/a) hb |
| (show Nat.log b (n/a)<k+1 by omega) |
| simpa only [pow_succ,Nat.mul_comm] using hh |
| have hNP : N<P := by nlinarith |
| have hNP' : (N:ℝ)<P := by exact_mod_cast hNP |
| have hPA : P*a≤n := (Nat.le_div_iff_mul_le ha).mp hP |
| have hnupper : n<(b*P)*a := (Nat.div_lt_iff_lt_mul ha).mp hP' |
| have hlow : (P:ℝ)^d*((P:ℝ)*f a)≤f n := by |
| have hh := (hgrow k).trans (hf hPA) |
| have he : (P:ℝ)^d*((P:ℝ)*f a) = ((b:ℝ)^(d+1))^k*f a := by |
| dsimp [P] |
| push_cast |
| rw [←mul_assoc,←pow_succ,←pow_mul,←pow_mul,Nat.mul_comm k (d+1)] |
| rw [he] |
| exact hh |
| have hupper : f n≤(P:ℝ)^d*(C*((b*a:ℕ):ℝ)^d) := by |
| have hnupper' : (n:ℝ)≤((b*P)*a:ℕ) := by exact_mod_cast hnupper.le |
| have hh := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (Nat.cast_nonneg _) hnupper' d) hC |
| apply hnpoly.trans |
| convert hh using 1 |
| push_cast |
| ring |
| have hPpos : (0:ℝ)<(P:ℝ)^d := by |
| apply pow_pos |
| exact_mod_cast (show 0<P by dsimp [P]; positivity) |
| have hsmall : C*((b*a:ℕ):ℝ)^d<(P:ℝ)*f a := |
| hNF.trans (mul_lt_mul_of_pos_right hNP' hfapos) |
| exact (not_lt_of_ge (hlow.trans hupper)) (mul_lt_mul_of_pos_left hsmall hPpos) |
|
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| lemma exists_local_ratio (f : ℕ → ℝ) (L : ℕ) (hL : 0<L) |
| (C : ℝ) (hC : 0<C) (hbound : f L≤C^L*f 0) : |
| ∃ i<L, f (i+1)≤C*f i := by |
| by_contra! hbad |
| have hgrow : ∀ i≤L, C^i*f 0≤f i := by |
| intro i hi |
| induction i with |
| | zero => simp |
| | succ i ih => |
| have hh := ih (by omega) |
| calc |
| _ = C*(C^i*f 0) := by rw [pow_succ]; ring |
| _ ≤ C*f i := mul_le_mul_of_nonneg_left hh hC.le |
| _ ≤ _ := (hbad i (by omega)).le |
| have hlast := hbad (L-1) (by omega) |
| rw [Nat.sub_add_cancel (by omega : 1≤L)] at hlast |
| have hg := mul_le_mul_of_nonneg_left (hgrow (L-1) (by omega)) hC.le |
| have he : C*(C^(L-1)*f 0)=C^L*f 0 := by |
| rw [←mul_assoc,←pow_succ',Nat.sub_add_cancel (by omega : 1≤L)] |
| rw [he] at hg |
| exact (not_lt_of_ge hbound) (hg.trans_lt hlast) |
|
|
| end CriticalPercolation.HarmonicAnalysis |
| end |
| end |
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| end OAI |
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