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f205175 | 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 | import Mathlib
namespace OAI
section
noncomputable section
open scoped Classical
namespace CriticalPercolation.HarmonicAnalysis
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)
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
end OAI
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