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0dd3af4 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | import Mathlib
namespace OAI
section
section
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
open scoped commutatorElement
section SourceDyadicArithmetic
open Classical Filter
open scoped Topology
noncomputable def sourceDyadicN (n : ℕ) : ℝ := (2:ℝ)^(4*n)
noncomputable def sourceDyadicWeight (n : ℕ) : ℝ := (Real.log (sourceDyadicN n))^(-3/4:ℝ)
def sourceDyadicIndices (n : ℕ) : Finset ℕ := Finset.Icc n (2*n)
theorem sourceDyadicN_pos (n : ℕ) : 0 < sourceDyadicN n := by unfold sourceDyadicN; positivity
theorem sourceDyadicN_one_le (n : ℕ) : 1 ≤ sourceDyadicN n := by
unfold sourceDyadicN
exact one_le_pow₀ (by norm_num)
theorem sourceDyadicN_fourth_root (n : ℕ) : (sourceDyadicN n)^(1/4:ℝ)=(2:ℝ)^n := by
unfold sourceDyadicN
rw [← Real.rpow_natCast_mul (by norm_num : (0:ℝ) ≤ 2)]
have he : ((4*n:ℕ):ℝ)*(1/4:ℝ)=(n:ℝ) := by push_cast; ring
rw [he,Real.rpow_natCast]
theorem sourceDyadicN_square_root (n : ℕ) : (sourceDyadicN n)^(1/2:ℝ)=(2:ℝ)^(2*n) := by
unfold sourceDyadicN
rw [← Real.rpow_natCast_mul (by norm_num : (0:ℝ) ≤ 2)]
have he : ((4*n:ℕ):ℝ)*(1/2:ℝ)=((2*n:ℕ):ℝ) := by push_cast; ring
rw [he,Real.rpow_natCast]
theorem sourceDyadicIndices_exact (n i : ℕ) :
i∈sourceDyadicIndices n ↔ (sourceDyadicN n)^(1/4:ℝ) ≤ (2:ℝ)^i ∧
(2:ℝ)^i ≤ (sourceDyadicN n)^(1/2:ℝ) := by
rw [sourceDyadicN_fourth_root,sourceDyadicN_square_root]
simp only [sourceDyadicIndices,Finset.mem_Icc,pow_le_pow_iff_right₀ (by norm_num : (1:ℝ) < 2)]
theorem sourceDyadicIndices_card (n : ℕ) : (sourceDyadicIndices n).card=n+1 := by
simp only [sourceDyadicIndices,Nat.card_Icc]
omega
theorem sourceDyadic_log (n : ℕ) : Real.log (sourceDyadicN n)=(4*Real.log 2)*(n:ℝ) := by
rw [sourceDyadicN,Real.log_pow]
push_cast
ring
theorem sourceDyadicWeight_square (n : ℕ) :
(sourceDyadicWeight n)^2=(4*Real.log 2)^(-3/2:ℝ)*(n:ℝ)^(-3/2:ℝ) := by
have hb : 0 ≤ 4*Real.log 2 := by positivity
unfold sourceDyadicWeight
rw [sourceDyadic_log,← Real.rpow_mul_natCast (mul_nonneg hb (Nat.cast_nonneg n))]
norm_num
rw [Real.mul_rpow hb (Nat.cast_nonneg n)]
theorem sourceDyadicWeight_energy_bound {n : ℕ} (hn : 1 ≤ n) :
((sourceDyadicIndices n).card:ℝ)*(sourceDyadicWeight n)^2 ≤ 2*(4*Real.log 2)^(-3/2:ℝ)*(n:ℝ)^(-1/2:ℝ) := by
have hn₀ : 0 < (n:ℝ) := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hn)
have hn₁ : (1:ℝ) ≤ n := by exact_mod_cast hn
rw [sourceDyadicIndices_card,Nat.cast_add,Nat.cast_one,sourceDyadicWeight_square]
have hc : 0 ≤ (4*Real.log 2)^(-3/2:ℝ)*(n:ℝ)^(-3/2:ℝ) := by positivity
have he : (n:ℝ)*(n:ℝ)^(-3/2:ℝ)=(n:ℝ)^(-1/2:ℝ) := by
conv_lhs => lhs; rw [← Real.rpow_one (n:ℝ)]
rw [← Real.rpow_add hn₀]
norm_num
calc
_ ≤ (2*(n:ℝ))*((4*Real.log 2)^(-3/2:ℝ)*(n:ℝ)^(-3/2:ℝ)) :=
mul_le_mul_of_nonneg_right (by linarith) hc
_ = _ := by rw [show (2*(n:ℝ))*((4*Real.log 2)^(-3/2:ℝ)*(n:ℝ)^(-3/2:ℝ))=
(2*(4*Real.log 2)^(-3/2:ℝ))*((n:ℝ)*(n:ℝ)^(-3/2:ℝ)) by ring,he]
theorem sourceDyadicWeight_energy_tendsto :
Tendsto (fun n => ((sourceDyadicIndices n).card:ℝ)*(sourceDyadicWeight n)^2) atTop (nhds 0) := by
have ht : Tendsto (fun n : ℕ => (n:ℝ)^(-1/2:ℝ)) atTop (nhds 0) := by
convert (tendsto_rpow_neg_atTop (show (0:ℝ) < 1/2 by norm_num)).comp
(tendsto_natCast_atTop_atTop (R:=ℝ)) using 1
norm_num
rfl
have ht' := ht.const_mul (2*(4*Real.log 2)^(-3/2:ℝ))
simp only [mul_zero] at ht'
apply squeeze_zero' (Eventually.of_forall (fun n => mul_nonneg (Nat.cast_nonneg _) (sq_nonneg _))) _ ht'
filter_upwards [eventually_ge_atTop 1] with n hn
exact sourceDyadicWeight_energy_bound hn
theorem sourceDyadic_fine_eventually (R : ℝ) {κ : ℝ} (hκ : 0 < κ) :
∀ᶠn : ℕ in atTop,∀i∈sourceDyadicIndices n,R ≤ κ/((2:ℝ)^i/sourceDyadicN n) := by
have ht := (tendsto_pow_atTop_atTop_of_one_lt (show (1:ℝ) < 4 by norm_num)).const_mul_atTop hκ
filter_upwards [ht.eventually_ge_atTop R] with n hn i hi
have hi' : i ≤ 2*n := (Finset.mem_Icc.mp hi).2
have hp : (2:ℝ)^i ≤ (2:ℝ)^(2*n) := pow_le_pow_right₀ (by norm_num) hi'
have he : κ/((2:ℝ)^(2*n)/sourceDyadicN n)=κ*(4:ℝ)^n := by
unfold sourceDyadicN
have hN : (2:ℝ)^(4*n)=((2:ℝ)^(2*n))^2 := by rw [← pow_mul]; congr 1; omega
have h4 : (4:ℝ)^n=(2:ℝ)^(2*n) := by rw [pow_mul]; norm_num
rw [hN,h4]
field_simp
calc
R ≤ κ*(4:ℝ)^n := hn
_ = κ/((2:ℝ)^(2*n)/sourceDyadicN n) := he.symm
_ ≤ _ := by
apply div_le_div_of_nonneg_left hκ.le (div_pos (by positivity) (sourceDyadicN_pos n))
exact div_le_div_of_nonneg_right hp (sourceDyadicN_pos n).le
end SourceDyadicArithmetic
end SimpleAmenable
end
end
end OAI
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