| import OAI.NumberTheory.TwoPoint.ShortIntervals.MRTCharacterZeroCount |
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|
| namespace OAI |
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| /-! A local analytic disk estimate with an explicit logarithmic growth budget. |
| The statement is independent of the physical radius, so it can be used on the |
| shrinking disks required by a Vinogradov--Korobov growth estimate. -/ |
|
|
| namespace TwoPointCorrelations |
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|
| open Complex Filter |
| open scoped BigOperators Classical Topology |
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|
| noncomputable def mrtDiskZeros (f : ℂ → ℂ) : Set ℂ := |
| Erdos970.zerosetKfR (7 / 8) (by norm_num) f |
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| lemma mrtDiskZeros_finite (f : ℂ → ℂ) |
| (hf : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, AnalyticAt ℂ f z) |
| (h0 : f 0 = 1) : (mrtDiskZeros f).Finite := by |
| apply Erdos970.lem_Contra_finiteKR (7 / 8) (by norm_num) (by norm_num) f hf |
| exact ⟨0, by simp, by rw [h0]; exact one_ne_zero⟩ |
|
|
| private lemma mrt_disk_factorization (f : ℂ → ℂ) |
| (hf : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, AnalyticAt ℂ f z) |
| (h0 : f 0 = 1) : |
| ∃ g : ℂ → ℂ → ℂ, ∀ ρ ∈ mrtDiskZeros f, |
| AnalyticAt ℂ (g ρ) ρ ∧ g ρ ρ ≠ 0 ∧ |
| ∀ᶠ w in 𝓝 ρ, f w = (w - ρ) ^ (analyticOrderAt f ρ).toNat * g ρ w := by |
| have hfactor : ∀ ρ : ℂ, ∃ g : ℂ → ℂ, AnalyticAt ℂ g ρ ∧ g ρ ≠ 0 ∧ |
| (ρ ∈ mrtDiskZeros f → |
| ∀ᶠ w in 𝓝 ρ, f w = (w - ρ) ^ (analyticOrderAt f ρ).toNat * g w) := by |
| intro ρ |
| by_cases hρ : ρ ∈ mrtDiskZeros f |
| · obtain ⟨g, hg, hgne, heq⟩ := Erdos970.lem_analytic_zero_factor |
| (15 / 16) (7 / 8) (by norm_num) (by norm_num) (by norm_num) f hf |
| (by rw [h0]; exact one_ne_zero) ρ hρ |
| exact ⟨g, hg, hgne, fun _ => heq⟩ |
| · exact ⟨fun _ => 1, analyticAt_const, one_ne_zero, fun h => (hρ h).elim⟩ |
| exact ⟨fun ρ => Classical.choose (hfactor ρ), fun ρ hρ => |
| ⟨(Classical.choose_spec (hfactor ρ)).1, |
| (Classical.choose_spec (hfactor ρ)).2.1, |
| (Classical.choose_spec (hfactor ρ)).2.2 hρ⟩⟩ |
|
|
| /-- Logarithmic derivative equals the actual finite zero sum with an error |
| linear in the logarithmic growth budget. -/ |
| theorem mrt_disk_logderiv (f : ℂ → ℂ) |
| (hf : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, AnalyticAt ℂ f z) |
| (h0 : f 0 = 1) {B : ℝ} (hB : 0 < B) |
| (hbound : ∀ z ∈ Metric.closedBall (0 : ℂ) (15 / 16), ‖f z‖ ≤ Real.exp B) |
| {z : ℂ} (hz : ‖z‖ ≤ 3 / 4) (hn : f z ≠ 0) : |
| ‖deriv f z / f z - |
| ∑ ρ ∈ (mrtDiskZeros_finite f hf h0).toFinset, |
| (analyticOrderAt f ρ).toNat / (z - ρ)‖ ≤ |
| mrtCharacterLogDerivativeConstant * B := by |
| obtain ⟨g, hg⟩ := mrt_disk_factorization f hf h0 |
| have hz' : z ∈ Metric.closedBall (0 : ℂ) (3 / 4 : ℝ) \ |
| Erdos970.zerosetKfR (7 / 8) (by norm_num) f := |
| ⟨by simpa using hz, fun h => hn h.2⟩ |
| have h := Erdos970.final_ineq1 (Real.exp B) (Real.one_lt_exp_iff.mpr hB) |
| (3 / 4) (4 / 5) (15 / 16) (7 / 8) |
| (by norm_num) (by norm_num) (by norm_num) (by norm_num) (by norm_num) |
| f hf h0 (mrtDiskZeros_finite f hf h0) hg hbound z hz' |
| simpa only [mrtDiskZeros, Real.log_exp, mrtCharacterLogDerivativeConstant] using h |
|
|
| /-- The total multiplicity in the inner disk has the same logarithmic cost. -/ |
| theorem mrt_disk_zero_count (f : ℂ → ℂ) |
| (hf : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, AnalyticAt ℂ f z) |
| (h0 : f 0 = 1) {B : ℝ} (hB : 0 < B) |
| (hbound : ∀ z ∈ Metric.closedBall (0 : ℂ) (15 / 16), ‖f z‖ ≤ Real.exp B) : |
| (∑ ρ ∈ (mrtDiskZeros_finite f hf h0).toFinset, |
| ((analyticOrderAt f ρ).toNat : ℝ)) ≤ |
| (1 / Real.log ((15 / 16 : ℝ) / (7 / 8))) * B := by |
| obtain ⟨g, hg⟩ := mrt_disk_factorization f hf h0 |
| have h := Erdos970.lem_sum_m_rho_bound (Real.exp B) (15 / 16) (7 / 8) |
| (Real.one_lt_exp_iff.mpr hB) (by norm_num) (by norm_num) (by norm_num) f hf |
| (by rw [h0]; exact one_ne_zero) h0 (mrtDiskZeros_finite f hf h0) g (fun z hz => hbound z (by simpa using hz)) hg |
| simpa only [mrtDiskZeros, Real.log_exp] using h |
|
|
| /-- A zero separation estimate completes the local norm bound. -/ |
| theorem mrt_disk_logderiv_norm (f : ℂ → ℂ) |
| (hf : ∀ z ∈ Metric.closedBall (0 : ℂ) 1, AnalyticAt ℂ f z) |
| (h0 : f 0 = 1) {B : ℝ} (hB : 0 < B) |
| (hbound : ∀ z ∈ Metric.closedBall (0 : ℂ) (15 / 16), ‖f z‖ ≤ Real.exp B) |
| {z : ℂ} (hz : ‖z‖ ≤ 3 / 4) (hn : f z ≠ 0) |
| {η : ℝ} (hη : 0 < η) |
| (hd : ∀ ρ ∈ mrtDiskZeros f, η ≤ ‖z - ρ‖) : |
| ‖deriv f z / f z‖ ≤ |
| (mrtCharacterLogDerivativeConstant + |
| (1 / Real.log ((15 / 16 : ℝ) / (7 / 8))) / η) * B := by |
| let S := (mrtDiskZeros_finite f hf h0).toFinset |
| let m : ℂ → ℕ := fun ρ => (analyticOrderAt f ρ).toNat |
| have hs : ‖∑ ρ ∈ S, (m ρ : ℂ) / (z - ρ)‖ ≤ (∑ ρ ∈ S, (m ρ : ℝ)) / η := by |
| calc |
| _ ≤ ∑ ρ ∈ S, ‖(m ρ : ℂ) / (z - ρ)‖ := norm_sum_le _ _ |
| _ ≤ ∑ ρ ∈ S, (m ρ : ℝ) / η := by |
| apply Finset.sum_le_sum |
| intro ρ hρ |
| rw [norm_div, Complex.norm_natCast] |
| exact div_le_div_of_nonneg_left (Nat.cast_nonneg _) hη |
| (hd ρ ((mrtDiskZeros_finite f hf h0).mem_toFinset.mp hρ)) |
| _ = _ := by rw [Finset.sum_div] |
| have hc := div_le_div_of_nonneg_right (mrt_disk_zero_count f hf h0 hB hbound) hη.le |
| have he := mrt_disk_logderiv f hf h0 hB hbound hz hn |
| have ht := norm_add_le |
| (deriv f z / f z - ∑ ρ ∈ S, (m ρ : ℂ) / (z - ρ)) |
| (∑ ρ ∈ S, (m ρ : ℂ) / (z - ρ)) |
| rw [sub_add_cancel] at ht |
| calc |
| _ ≤ mrtCharacterLogDerivativeConstant * B + |
| ((1 / Real.log ((15 / 16 : ℝ) / (7 / 8))) * B) / η := |
| ht.trans (add_le_add he (hs.trans hc)) |
| _ = _ := by ring |
|
|
| end TwoPointCorrelations |
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| end OAI |
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