| import OAI.Dynamics.StandardMap.BridgeEligibility |
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| namespace OAI |
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| open MeasureTheory Set |
| open scoped ENNReal BigOperators |
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| 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 |
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| 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 |
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| end OAI |
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