File size: 3,948 Bytes
92416a2 | 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 | import OAI.Dynamics.StandardMap.BridgeEligibility
namespace OAI
open MeasureTheory Set
open scoped ENNReal BigOperators
open MeasureTheory Set Filter Metric
open scoped Topology ENNReal
namespace StandardMapEntropy
lemma eligible_dirichlet_bound (v : ℕ → ℝ) (M D : ℝ) (B J : ℕ)
(hM : 1 < M) (hJ : 1 ≤ J) (hJB : J ≤ B+1)
(hD : M^((49/50:ℝ)*(B:ℝ)) ≤ D)
(hv : ∀ p, 1 ≤ p → p ≤ B → |v p|+1 ≤ M)
(s : ℂ) (hs : ‖s‖=1) (hver : |s.re| ≤ 1/4)
(hpref : ∀ p, 1 ≤ p → p ≤ B → ‖transferProduct v p s‖ ≤ 3*M^(-(47/50:ℝ)*(p:ℝ)))
(row : ℂ →L[ℝ] ℝ) (hroweq : ∀ z, row z=linearSolution v z.im z.re J)
(hsmall : |row s| ≤ D⁻¹) (hrow : D/3 ≤ |row 1|) :
tSolution v J ≠ 0 ∧ ∀ p, 1 ≤ p → p < J →
|dirichletSolution v J p| ≤ 12*M^(-(9/10:ℝ)*(p:ℝ)) := by
have hMp : 0 < M := by linarith
have hDp : 0 < D := (Real.rpow_pos_of_pos hMp _).trans_le hD
have ht : tSolution v J ≠ 0 := by
have hr : row 1≠0 := abs_pos.mp (by linarith : 0 < |row 1|)
simpa only [hroweq,Complex.one_im,Complex.one_re,tSolution] using hr
obtain ⟨u,e,hui,hru,heq,he⟩:=normalized_kernel_from_stable row s D hDp hs hver hrow hsmall
have huj : linearSolution v 1 u.re J=0 := by simpa only [hroweq,hui] using hru
have heqU :=initial_kernel_dirichlet v J u hui hJ huj ht
refine ⟨ht,?_⟩
intro p hp hpJ
have hpB : p ≤ B := by omega
have hup : ‖transferProduct v p‖ ≤ M^(p:ℝ) := by
simpa only [Real.rpow_natCast] using (transferProduct_norm_bounds v M p (by linarith)
(fun i hi hip => hv i hi (by omega))).2
have hc:=eligible_row_contraction (transferProduct v) s u e M D B p hp hpB hM hs hver heq he
hD (hpref p hp hpB) hup
have hu3 : (⟨u.re,1⟩:ℂ)=u := by apply Complex.ext <;> simp [hui]
have he : (transferProduct v p u).im=dirichletSolution v J p := by
rw [← hu3,transferProduct_pair]
exact heqU p
calc
_ = |(transferProduct v p u).im| := congrArg abs he.symm
_ ≤ ‖transferProduct v p u‖ := Complex.abs_im_le_norm _
_ ≤ 12*M^(-(47/50:ℝ)*(p:ℝ)) := hc
_ ≤ _ := by
apply mul_le_mul_of_nonneg_left (Real.rpow_le_rpow_of_exponent_le hM.le ?_) (by norm_num)
have : (0:ℝ) ≤ p := Nat.cast_nonneg _
linarith
lemma fast_prefix_dirichlet (v : ℕ → ℝ) (M : ℝ) (B : ℕ)
(hB : 2 ≤ B) (hM : 1 < M) (hq : M^(-(49/25:ℝ)) ≤ 1/2)
(hsmall : 3*M^(-(47/50:ℝ)) ≤ 1/4)
(hv : ∀ i, 1 ≤ i → i ≤ B → |v i|+1 ≤ M)
(hg : ∀ i, 1 ≤ i → i ≤ B → M^((49/50:ℝ)*(i:ℝ)) ≤ ‖transferProduct v i‖) :
∃ J : ℕ, (J=B ∨ J=B+1) ∧ tSolution v J ≠ 0 ∧
∀ p, 1 ≤ p → p < J → |dirichletSolution v J p| ≤ 12*M^(-(9/10:ℝ)*(p:ℝ)) := by
obtain ⟨s,hu,hver,hs,hpref⟩:=prefix_stable_vertical v M B (by omega) hM hq hsmall hv hg
have helig:=(transferProduct_area v B).eligible_row s hu hver hs
rcases helig with hleft|hright
· refine ⟨B+1,Or.inr rfl,?_⟩
apply eligible_dirichlet_bound v M ‖transferProduct v B‖ B (B+1) hM (by omega) le_rfl
(hg B (by omega) le_rfl) hv s hu hver hpref (Complex.reCLM.comp (transferProduct v B)) ?_ ?_ hleft
· intro z
change (transferProduct v B z).re=linearSolution v z.im z.re (B+1)
have hz : (⟨z.re,z.im⟩:ℂ)=z := rfl
rw [← hz,transferProduct_pair]
· exact (Complex.abs_re_le_norm _).trans_eq hs
· refine ⟨B,Or.inl rfl,?_⟩
apply eligible_dirichlet_bound v M ‖transferProduct v B‖ B B hM (by omega) (by omega)
(hg B (by omega) le_rfl) hv s hu hver hpref (Complex.imCLM.comp (transferProduct v B)) ?_ ?_ hright
· intro z
change (transferProduct v B z).im=linearSolution v z.im z.re B
have hz : (⟨z.re,z.im⟩:ℂ)=z := rfl
rw [← hz,transferProduct_pair]
· exact (Complex.abs_im_le_norm _).trans_eq hs
end StandardMapEntropy
end OAI
|