| import OAI.Analysis.IntegralMeans.TraceIdentity |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory Filter Function InnerProductSpace |
| open scoped Topology ComplexConjugate Manifold NNReal ENNReal InnerProductSpace Classical |
| open MeasureTheory Function |
| open Set Filter |
| open Set MeasureTheory Filter Function |
| open Set MeasureTheory Filter Function InnerProductSpace |
| open TopologicalSpace |
| open scoped CompactlySupported |
| open scoped ENNReal |
| open scoped Manifold |
| open scoped Topology CompactlySupported ComplexConjugate |
| open scoped Topology ComplexConjugate Manifold NNReal ENNReal InnerProductSpace Classical |
| open scoped Topology ENNReal NNReal |
| namespace Brennan |
|
|
| attribute [local irreducible] classWeight |
| attribute [local irreducible] classFun |
|
|
| lemma halfPlane_ball_subset {z : ℂ} (_hz : z ∈ halfPlane) : |
| Metric.ball z z.im ⊆ halfPlane := by |
| intro w hw |
| have hh : ‖w-z‖ < z.im := by simpa only [Metric.mem_ball,dist_eq_norm] using hw |
| have hb := Complex.abs_im_le_norm (w-z) |
| change 0 < w.im |
| simp only [Complex.sub_im] at hb |
| linarith [neg_le_abs (w.im-z.im)] |
|
|
| lemma omitted_quarter_bound {F : ℂ → ℂ} (hf : UnivalentOn F halfPlane) |
| {z ξ : ℂ} (hz : z ∈ halfPlane) (hξ : ξ ∉ F '' halfPlane) : |
| z.im*‖deriv F z‖ ≤ 4*‖F z-ξ‖ := by |
| have hr : 0 < z.im := hz |
| have hb := univalent_ball_quarter hr |
| (show UnivalentOn F (Metric.ball z z.im) from ⟨hf.1.mono (halfPlane_ball_subset hz),hf.2.mono (halfPlane_ball_subset hz)⟩) |
| have hn : ξ ∉ Metric.ball (F z) (z.im*‖deriv F z‖/4) := by |
| intro hh |
| exact hξ ((image_mono (halfPlane_ball_subset hz)) (hb hh)) |
| have hn' : z.im*‖deriv F z‖/4 ≤ ‖F z-ξ‖ := by |
| simpa only [Metric.mem_ball,dist_eq_norm,not_lt,norm_sub_rev] using hn |
| linarith |
| |
| lemma class_omitted_cell_bound {M : ℝ} (hM : 0 ≤ M) |
| (hb : ∀ (g : DiskClass) (w : ℂ), |w.re| ≤ 1 → (1/2 : ℝ) ≤ w.im → w.im ≤ 1 → ‖classFun g w‖ ≤ M) |
| (g : DiskClass) {h x y x' : ℝ} {ξ : ℂ} (hh : 0 < h) |
| (hy : h ≤ y) (hyy : y ≤ 2*h) (hxx : |x'-x| ≤ h) |
| (hξ : ξ ∉ classFun g '' halfPlane) : |
| ‖classFun g (x'+h*Complex.I)-ξ‖ ≤ (1+4*M)*‖classFun g (x+y*Complex.I)-ξ‖ := by |
| let z : ℂ := x+y*Complex.I |
| let w : ℂ := ((x'-x)/y : ℝ)+(h/y : ℝ)*Complex.I |
| have hy0 : 0 < y := hh.trans_le hy |
| have hz : z ∈ halfPlane := by simpa [z,halfPlane] using hy0 |
| have hwy : w.im = h/y := by simp [w] |
| have hwx : w.re = (x'-x)/y := by simp [w] |
| have hw : w ∈ halfPlane := by simpa [halfPlane,hwy] using (div_pos hh hy0) |
| have hwr : |w.re| ≤ 1 := by rw [hwx,abs_div,abs_of_pos hy0]; exact (div_le_one hy0).mpr (hxx.trans hy) |
| have hwl : (1/2 : ℝ) ≤ w.im := by rw [hwy,le_div_iff₀ hy0]; linarith |
| have hwu : w.im ≤ 1 := by rw [hwy]; exact (div_le_one hy0).mpr hy |
| have hwa : affine z w = x'+h*Complex.I := by |
| apply Complex.ext <;> simp [affine,z,w,Complex.mul_re,Complex.mul_im] |
| · field_simp |
| ring |
| · field_simp |
| have hd : ‖deriv (classFun g) z‖ > 0 := norm_pos_iff.mpr (univalent_deriv_ne_zero isOpen_halfPlane (classFun_schlicht g).1 hz) |
| have hm := hb (rerootClass g ⟨z,hz⟩) w hwr hwl hwu |
| rw [classFun_rerootClass _ _ hw,reroot,hwa,norm_div,norm_mul,Complex.norm_real] at hm |
| have hzim : z.im = y := by simp [z] |
| rw [hzim,Real.norm_eq_abs,abs_of_pos hy0] at hm |
| have hm' := (div_le_iff₀ (mul_pos hy0 hd)).mp hm |
| have hq := omitted_quarter_bound (classFun_schlicht g).1 hz hξ |
| rw [hzim] at hq |
| have hsum := norm_add_le (classFun g (x'+h*Complex.I)-classFun g z) (classFun g z-ξ) |
| simp only [sub_add_sub_cancel] at hsum |
| have hhq := mul_le_mul_of_nonneg_left hq hM |
| dsimp only [z] at * |
| nlinarith |
| |
| def closedPositiveSuperlevel (F : ℂ → ℂ) (k δ : ℝ) (ξ : ℂ) : Set ℂ := |
| {z | 0 < z.im ∧ ‖F z-ξ‖ ≤ z.im^k/δ} |
| |
| lemma interior_positive_superlevel_height {F : ℂ → ℂ} {k δ : ℝ} |
| (hk : 0 < k) (_hδ : 0 < δ) |
| (hi : InjOn F {z : ℂ | 0 < z.im}) |
| (hopen : ∀ z : ℂ, 0 < z.im → map F (𝓝 z) = 𝓝 (F z)) |
| {ξ : ℂ} (hξ : ξ ∈ F '' {z : ℂ | 0 < z.im}) : |
| ∃ η : ℝ, 0 < η ∧ ∀ z : ℂ, 0 < z.im → |
| ‖F z-ξ‖ ≤ z.im^k/δ → η ≤ z.im := by |
| obtain ⟨z₀,hz₀,rfl⟩ := hξ |
| change 0 < z₀.im at hz₀ |
| have hn : {z : ℂ | z₀.im/2 < z.im} ∈ 𝓝 z₀ := |
| (isOpen_lt continuous_const Complex.continuous_im).mem_nhds (by change z₀.im/2 < z₀.im; linarith) |
| have hn' := image_mem_map (m := F) hn |
| rw [hopen z₀ hz₀] at hn' |
| obtain ⟨r,hr,hrB⟩ := Metric.mem_nhds_iff.mp hn' |
| have hc : ContinuousAt (fun y : ℝ => y^k/δ) 0 := |
| (Real.continuous_rpow_const hk.le).continuousAt.div_const δ |
| have he : (0 : ℝ)^k/δ = 0 := by rw [Real.zero_rpow (ne_of_gt hk),zero_div] |
| have hevent : ∀ᶠ y : ℝ in 𝓝 0, y^k/δ < r := by |
| have hh := hc.eventually (gt_mem_nhds (show (0 : ℝ)^k/δ < r by rwa [he])) |
| exact hh |
| obtain ⟨a,ha,haB⟩ := Metric.mem_nhds_iff.mp hevent |
| refine ⟨min a (z₀.im/2),lt_min ha (by linarith),fun z hz hzw => ?_⟩ |
| by_contra! hsmall |
| have hza : z.im < a := hsmall.trans_le (min_le_left _ _) |
| have hm : F z ∈ Metric.ball (F z₀) r := by |
| rw [Metric.mem_ball,dist_eq_norm] |
| exact hzw.trans_lt (haB (by simpa only [Metric.mem_ball,dist_zero_right,Real.norm_eq_abs,abs_of_pos hz] using hza)) |
| obtain ⟨w,hw,hwe⟩ := hrB hm |
| have hwH : 0 < w.im := by change z₀.im/2 < w.im at hw; linarith |
| have hwz : w = z := hi hwH hz hwe |
| subst w |
| change z₀.im/2 < z.im at hw |
| exact (not_lt_of_ge hw.le) (hsmall.trans_le (min_le_right _ _)) |
|
|
| lemma positive_superlevel_compact {F : ℂ → ℂ} {k δ : ℝ} {ξ : ℂ} |
| (hk : 0 ≤ k) (hδ : 0 < δ) |
| (hc : ContinuousOn F {z : ℂ | 0 < z.im}) |
| (ht : HasQuarticTail F (4*k)) |
| (hp : ∀ K : Set ℂ, IsCompact K → ∀ η : ℝ, 0 < η → |
| IsCompact ({z : ℂ | η ≤ z.im} ∩ F ⁻¹' K)) |
| (hheight : ∃ η : ℝ, 0 < η ∧ ∀ z : ℂ, 0 < z.im → |
| ‖F z-ξ‖ ≤ z.im^k/δ → η ≤ z.im) : |
| IsCompact (closedPositiveSuperlevel F k δ ξ) := by |
| obtain ⟨η,hη,hηb⟩ := hheight |
| obtain ⟨R₀,hR₀⟩ := (hp (Metric.closedBall (0 : ℂ) (2*‖ξ‖+1)) |
| (isCompact_closedBall _ _) η hη).isBounded.subset_closedBall (0 : ℂ) |
| obtain ⟨R₁,hR₁⟩ := quartic_tail_iff_power ht (δ/2) (by linarith) |
| have hb : closedPositiveSuperlevel F k δ ξ ⊆ Metric.closedBall 0 (max R₀ R₁) := by |
| intro z hz |
| rw [Metric.mem_closedBall,dist_zero_right] |
| by_contra! hn |
| have hno : 2*‖ξ‖+1 < ‖F z‖ := by |
| by_contra! hle |
| have hh := hR₀ ⟨hηb z hz.1 hz.2,by simpa using hle⟩ |
| have hh' : ‖z‖ ≤ R₀ := by simpa using hh |
| linarith [le_max_left R₀ R₁] |
| have hFn : 0 < ‖F z‖ := by nlinarith [norm_nonneg ξ] |
| have ht' := hR₁ z hz.1 ((le_max_right _ _).trans hn.le) |
| rw [← div_eq_mul_inv,div_lt_iff₀ hFn] at ht' |
| have hpot := (le_div_iff₀ hδ).mp hz.2 |
| have hnorm : ‖F z‖ ≤ ‖F z-ξ‖+‖ξ‖ := by |
| simpa only [sub_add_cancel] using norm_add_le (F z-ξ) ξ |
| nlinarith |
| have he : closedPositiveSuperlevel F k δ ξ = |
| {z : ℂ | η ≤ z.im} ∩ (fun z => ‖F z-ξ‖-z.im^k/δ) ⁻¹' Iic 0 := by |
| ext z |
| constructor |
| · intro hz |
| exact ⟨hηb z hz.1 hz.2,sub_nonpos.mpr hz.2⟩ |
| · intro hz |
| exact ⟨hη.trans_le hz.1,sub_nonpos.mp hz.2⟩ |
| have hcc : ContinuousOn (fun z : ℂ => ‖F z-ξ‖-z.im^k/δ) {z : ℂ | η ≤ z.im} := by |
| apply ((hc.mono (fun z hz => hη.trans_le hz)).sub continuousOn_const).norm.sub |
| exact ((Real.continuous_rpow_const hk).comp Complex.continuous_im).continuousOn.div_const δ |
| apply (isCompact_closedBall (0 : ℂ) (max R₀ R₁)).of_isClosed_subset _ hb |
| rw [he] |
| exact hcc.preimage_isClosed_of_isClosed (isClosed_le continuous_const Complex.continuous_im) isClosed_Iic |
| |
| def realCell (h : ℝ) (j : ℤ) : Set ℝ := Ico ((j : ℝ)*h) (((j : ℝ)+1)*h) |
| |
| lemma realCell_disjoint {h : ℝ} (hh : 0 < h) : Pairwise (Function.onFun Disjoint (realCell h)) := by |
| intro i j hij |
| apply Set.disjoint_left.mpr |
| intro x hx hy |
| rcases lt_or_gt_of_ne hij with hij | hji |
| · have hhij : (i : ℝ)+1 ≤ j := by exact_mod_cast Int.add_one_le_iff.mpr hij |
| have hm := mul_le_mul_of_nonneg_right hhij hh.le |
| exact (not_lt_of_ge (hm.trans hy.1)) hx.2 |
| · have hhji : (j : ℝ)+1 ≤ i := by exact_mod_cast Int.add_one_le_iff.mpr hji |
| have hm := mul_le_mul_of_nonneg_right hhji hh.le |
| exact (not_lt_of_ge (hm.trans hx.1)) hy.2 |
| |
| lemma measure_realCell {h : ℝ} (_hh : 0 ≤ h) (j : ℤ) : |
| volume (realCell h j) = ENNReal.ofReal h := by |
| rw [realCell,Real.volume_Ico] |
| congr 1 |
| ring |
| |
| lemma mem_realCell_floor {h : ℝ} (hh : 0 < h) (x : ℝ) : x ∈ realCell h ⌊x/h⌋ := by |
| constructor |
| · exact (le_div_iff₀ hh).mp (Int.floor_le _) |
| · exact (div_lt_iff₀ hh).mp (Int.lt_floor_add_one _) |
| |
| lemma realCell_length_bound {h : ℝ} (_hh : 0 < h) {j : ℤ} {x x' : ℝ} |
| (hx : x ∈ realCell h j) (hx' : x' ∈ realCell h j) : |x'-x| ≤ h := by |
| rw [abs_le] |
| constructor <;> dsimp [realCell] at hx hx' <;> nlinarith [hx.1,hx.2,hx'.1,hx'.2] |
|
|
| lemma cell_target_cover_bound {h r : ℝ} (hh : 0 < h) (hr : 0 ≤ r) |
| (W : Set ℤ) (centers : ℤ → ℂ) {S : Set ℝ} {C : ℝ≥0∞} |
| (hsub : ∀ j ∈ W, realCell h j ⊆ S) (hS : volume S ≤ C) : |
| volume (⋃ j : W, Metric.closedBall (centers j) r) ≤ |
| (ENNReal.ofReal (Real.pi*r^2)/ENNReal.ofReal h)*C := by |
| have hlen : (∑' j : W, ENNReal.ofReal h) ≤ C := by |
| have hd : Pairwise (Function.onFun Disjoint (fun j : W => realCell h j)) := by |
| intro i j hij |
| exact realCell_disjoint hh (fun he => hij (Subtype.ext he)) |
| have he := measure_iUnion (μ := volume) hd (fun _ => measurableSet_Ico) |
| simp only [measure_realCell hh.le] at he |
| rw [← he] |
| exact (measure_mono (iUnion_subset (fun (j : W) => hsub j.val j.property))).trans hS |
| have harea (j : W) : volume (Metric.closedBall (centers j) r) = ENNReal.ofReal (Real.pi*r^2) := by |
| rw [Complex.volume_closedBall] |
| have hpi : (NNReal.pi : ℝ≥0∞) = ENNReal.ofReal Real.pi := by |
| rw [ENNReal.ofReal,ENNReal.coe_inj] |
| apply Subtype.ext |
| exact (Real.coe_toNNReal Real.pi Real.pi_pos.le).symm |
| rw [hpi,ENNReal.ofReal_mul Real.pi_pos.le,ENNReal.ofReal_pow hr] |
| exact mul_comm _ _ |
| calc |
| _ ≤ ∑' j : W, volume (Metric.closedBall (centers j) r) := measure_iUnion_le _ |
| _ = ∑' j : W, (ENNReal.ofReal (Real.pi*r^2)/ENNReal.ofReal h)*ENNReal.ofReal h := by |
| apply tsum_congr |
| intro j |
| rw [harea] |
| exact (ENNReal.div_mul_cancel (ne_of_gt (ENNReal.ofReal_pos.mpr hh)) ENNReal.ofReal_ne_top).symm |
| _ = (ENNReal.ofReal (Real.pi*r^2)/ENNReal.ofReal h)*(∑' j : W, ENNReal.ofReal h) := ENNReal.tsum_mul_left |
| _ ≤ _ := mul_le_mul le_rfl hlen zero_le zero_le |
|
|
| lemma summable_geometric_ofReal {r : ℝ} (hr0 : 0 ≤ r) (hr1 : r < 1) : |
| (∑' n : ℕ, ENNReal.ofReal (r^n)) < ∞ := by |
| have hs : Summable (fun n : ℕ => (⟨r^n,pow_nonneg hr0 n⟩ : ℝ≥0)) := |
| NNReal.summable_coe.mp (summable_geometric_of_lt_one hr0 hr1) |
| have ht := ENNReal.tsum_coe_ne_top_iff_summable.mpr hs |
| apply lt_top_iff_ne_top.mpr |
| convert ht using 1 |
| congr 1 |
| ext n |
| exact ENNReal.ofReal_eq_coe_nnreal (pow_nonneg hr0 n) |
| |
| lemma ae_eventually_outside_geometric_covers {E : ℕ → Set ℂ} {C : ℝ≥0∞} (hC : C < ∞) |
| {r : ℝ} (hr0 : 0 ≤ r) (hr1 : r < 1) |
| (hE : ∀ n, volume (E n) ≤ C*ENNReal.ofReal (r^n)) : |
| ∀ᵐ ξ : ℂ, ∀ᶠ n : ℕ in atTop, ξ ∉ E n := by |
| apply ae_eventually_notMem |
| apply ne_top_of_le_ne_top _ (ENNReal.tsum_le_tsum hE) |
| rw [ENNReal.tsum_mul_left] |
| exact ne_of_lt (ENNReal.mul_lt_top hC (summable_geometric_ofReal hr0 hr1)) |
| |
| def dyadicHeight (n : ℕ) : ℝ := (1/2)^n |
| |
| lemma dyadicHeight_pos (n : ℕ) : 0 < dyadicHeight n := pow_pos (by norm_num) n |
| |
| lemma dyadicHeight_le_one (n : ℕ) : dyadicHeight n ≤ 1 := pow_le_one₀ (by norm_num) (by norm_num) |
| |
| lemma exists_dyadic_slab {y : ℝ} (hy : 0 < y) (hy1 : y ≤ 1) : |
| ∃ n : ℕ, dyadicHeight n ≤ y ∧ y ≤ 2*dyadicHeight n := by |
| obtain ⟨n,hn,hn'⟩ := exists_nat_pow_near_of_lt_one hy hy1 |
| (by norm_num : (0 : ℝ) < 1/2) (by norm_num : (1/2 : ℝ) < 1) |
| refine ⟨n+1,hn.le,?_⟩ |
| dsimp [dyadicHeight] |
| rw [pow_succ] |
| nlinarith |
|
|
| lemma dyadicHeight_antitone : Antitone dyadicHeight := by |
| intro m n hmn |
| exact pow_le_pow_of_le_one (by norm_num) (by norm_num) hmn |
|
|
| lemma small_height_dyadic_slab (N : ℕ) {y : ℝ} (hy : 0 < y) (hyN : y < dyadicHeight N) : |
| ∃ n ≥ N, dyadicHeight n ≤ y ∧ y ≤ 2*dyadicHeight n := by |
| obtain ⟨n,hn,hn'⟩ := exists_dyadic_slab hy (hyN.le.trans (dyadicHeight_le_one N)) |
| refine ⟨n,?_,hn,hn'⟩ |
| by_contra! h |
| exact (not_lt_of_ge ((dyadicHeight_antitone h.le).trans hn)) hyN |
|
|
| lemma slab_power_bound {k h y : ℝ} (hk : 1 ≤ k) (hh : 0 < h) (hh1 : h ≤ 1) |
| (hy : 0 ≤ y) (hyh : y ≤ 2*h) : y^k ≤ (2 : ℝ)^k*h := by |
| calc |
| y^k ≤ (2*h)^k := Real.rpow_le_rpow hy hyh (by linarith) |
| _ = (2 : ℝ)^k*h^k := Real.mul_rpow (by norm_num) hh.le |
| _ ≤ _ := mul_le_mul_of_nonneg_left (Real.rpow_le_self_of_le_one hh.le hh1 hk) (by positivity) |
|
|
| def omittedTargets (F : ℂ → ℂ) (k R δ h : ℝ) : Set ℂ := |
| {ξ | ξ ∉ F '' {z : ℂ | 0 < z.im} ∧ ‖ξ‖ ≤ R ∧ |
| ∃ x y : ℝ, h ≤ y ∧ y ≤ 2*h ∧ ‖F (x+y*Complex.I)-ξ‖ ≤ y^k/δ} |
|
|
| lemma omitted_targets_cover {F : ℂ → ℂ} {k L R δ : ℝ} |
| (hk : 1 ≤ k) (hL : 0 ≤ L) (hδ : 0 < δ) |
| (hcell : ∀ h x y x' : ℝ, ∀ ξ : ℂ, 0 < h → h ≤ y → y ≤ 2*h → |x'-x| ≤ h → |
| ξ ∉ F '' {z : ℂ | 0 < z.im} → |
| ‖F (x'+h*Complex.I)-ξ‖ ≤ L*‖F (x+y*Complex.I)-ξ‖) |
| (htrace : ∀ K : Set ℂ, IsCompact K → ∃ C : ℝ≥0∞, C < ∞ ∧ ∀ ε : ℝ, 0 < ε → |
| volume {x : ℝ | F (x+ε*Complex.I) ∈ K} ≤ C) : |
| ∃ C : ℝ≥0∞, C < ∞ ∧ ∀ h : ℝ, 0 < h → h ≤ 1 → |
| volume (omittedTargets F k R δ h) ≤ C*ENNReal.ofReal h := by |
| let D := L*(2 : ℝ)^k/δ |
| have hD : 0 ≤ D := div_nonneg (mul_nonneg hL (by positivity)) hδ.le |
| let K := Metric.closedBall (0 : ℂ) (R+D) |
| obtain ⟨C,hC,hCt⟩ := htrace K (isCompact_closedBall _ _) |
| refine ⟨ENNReal.ofReal (Real.pi*D^2)*C, ENNReal.mul_lt_top ENNReal.ofReal_lt_top hC,fun h hh hh1 => ?_⟩ |
| let W : Set ℤ := {j | ∃ ξ ∈ omittedTargets F k R δ h, |
| ∃ x y : ℝ, x ∈ realCell h j ∧ h ≤ y ∧ y ≤ 2*h ∧ ‖F (x+y*Complex.I)-ξ‖ ≤ y^k/δ} |
| let center (j : ℤ) := F ((((j : ℝ)+1/2)*h : ℝ)+h*Complex.I) |
| have hb {j : ℤ} {ξ : ℂ} (hξ : ξ ∈ omittedTargets F k R δ h) |
| {x y x' : ℝ} (hx : x ∈ realCell h j) (hy : h ≤ y) (hyy : y ≤ 2*h) |
| (hFw : ‖F (x+y*Complex.I)-ξ‖ ≤ y^k/δ) (hx' : x' ∈ realCell h j) : |
| ‖F (x'+h*Complex.I)-ξ‖ ≤ D*h := by |
| have ht := hcell h x y x' ξ hh hy hyy (realCell_length_bound hh hx hx') hξ.1 |
| have hp := div_le_div_of_nonneg_right (slab_power_bound hk hh hh1 (hh.le.trans hy) hyy) hδ.le |
| have hp' := mul_le_mul_of_nonneg_left (hFw.trans hp) hL |
| dsimp [D] |
| calc |
| _ ≤ L*((2 : ℝ)^k*h/δ) := ht.trans hp' |
| _ = _ := by ring |
| have hsub (j : ℤ) (hj : j ∈ W) : realCell h j ⊆ {x : ℝ | F (x+h*Complex.I) ∈ K} := by |
| rcases hj with ⟨ξ,hξ,x,y,hx,hy,hyy,hFw⟩ |
| intro x' hx' |
| have hhB := hb hξ hx hy hyy hFw hx' |
| have hn := norm_add_le (F (x'+h*Complex.I)-ξ) ξ |
| simp only [sub_add_cancel] at hn |
| change F (x'+h*Complex.I) ∈ Metric.closedBall (0 : ℂ) (R+D) |
| rw [Metric.mem_closedBall,dist_zero_right] |
| nlinarith [hξ.2.1,mul_le_of_le_one_right hD hh1] |
| have hcovered : omittedTargets F k R δ h ⊆ ⋃ j : W, Metric.closedBall (center j) (D*h) := by |
| intro ξ hξ |
| obtain ⟨x,y,hy,hyy,hFw⟩ := hξ.2.2 |
| let j : ℤ := ⌊x/h⌋ |
| have hx : x ∈ realCell h j := mem_realCell_floor hh _ |
| have hj : j ∈ W := ⟨ξ,hξ,x,y,hx,hy,hyy,hFw⟩ |
| apply mem_iUnion.mpr |
| refine ⟨⟨j,hj⟩,?_⟩ |
| rw [Metric.mem_closedBall,dist_eq_norm,norm_sub_rev] |
| change ‖F ((((j : ℝ)+1/2)*h : ℝ)+h*Complex.I)-ξ‖ ≤ D*h |
| exact hb (x' := ((j : ℝ)+1/2)*h) hξ hx hy hyy hFw (by constructor <;> nlinarith) |
| calc |
| _ ≤ volume (⋃ j : W, Metric.closedBall (center j) (D*h)) := measure_mono hcovered |
| _ ≤ (ENNReal.ofReal (Real.pi*(D*h)^2)/ENNReal.ofReal h)*C := |
| cell_target_cover_bound hh (mul_nonneg hD hh.le) W center hsub (hCt h hh) |
| _ = _ := by |
| rw [← ENNReal.ofReal_div_of_pos hh] |
| have he : Real.pi*(D*h)^2/h = (Real.pi*D^2)*h := by field_simp |
| rw [he,ENNReal.ofReal_mul (by positivity)] |
| ring |
| |
| lemma ae_omitted_positive_superlevel_height {F : ℂ → ℂ} {k L R δ : ℝ} |
| (hk : 1 ≤ k) (hL : 0 ≤ L) (hδ : 0 < δ) |
| (hcell : ∀ h x y x' : ℝ, ∀ ξ : ℂ, 0 < h → h ≤ y → y ≤ 2*h → |x'-x| ≤ h → |
| ξ ∉ F '' {z : ℂ | 0 < z.im} → |
| ‖F (x'+h*Complex.I)-ξ‖ ≤ L*‖F (x+y*Complex.I)-ξ‖) |
| (htrace : ∀ K : Set ℂ, IsCompact K → ∃ C : ℝ≥0∞, C < ∞ ∧ ∀ ε : ℝ, 0 < ε → |
| volume {x : ℝ | F (x+ε*Complex.I) ∈ K} ≤ C) : |
| ∀ᵐ ξ : ℂ, ξ ∉ F '' {z : ℂ | 0 < z.im} → ‖ξ‖ ≤ R → |
| ∃ η : ℝ, 0 < η ∧ ∀ z : ℂ, 0 < z.im → |
| ‖F z-ξ‖ ≤ z.im^k/δ → η ≤ z.im := by |
| obtain ⟨C,hC,hCb⟩ := omitted_targets_cover hk hL hδ hcell htrace (R := R) |
| have hae := ae_eventually_outside_geometric_covers hC |
| (by norm_num : (0 : ℝ) ≤ 1/2) (by norm_num : (1/2 : ℝ) < 1) |
| (E := fun n => omittedTargets F k R δ (dyadicHeight n)) |
| (fun n => hCb _ (dyadicHeight_pos n) (dyadicHeight_le_one n)) |
| filter_upwards [hae] with ξ hξ |
| intro ho hR |
| obtain ⟨N,hN⟩ := eventually_atTop.mp hξ |
| refine ⟨dyadicHeight N,dyadicHeight_pos N,fun z hz hzw => ?_⟩ |
| by_contra! hh |
| obtain ⟨n,hn,hy,hyy⟩ := small_height_dyadic_slab N hz hh |
| apply hN n hn |
| refine ⟨ho,hR,z.re,z.im,hy,hyy,?_⟩ |
| simpa only [Complex.re_add_im] using hzw |
|
|
| lemma ae_positive_superlevel_height {F : ℂ → ℂ} {k L : ℝ} |
| (hk : 1 ≤ k) (hL : 0 ≤ L) |
| (hi : InjOn F {z : ℂ | 0 < z.im}) |
| (hopen : ∀ z : ℂ, 0 < z.im → map F (𝓝 z) = 𝓝 (F z)) |
| (hcell : ∀ h x y x' : ℝ, ∀ ξ : ℂ, 0 < h → h ≤ y → y ≤ 2*h → |x'-x| ≤ h → |
| ξ ∉ F '' {z : ℂ | 0 < z.im} → |
| ‖F (x'+h*Complex.I)-ξ‖ ≤ L*‖F (x+y*Complex.I)-ξ‖) |
| (htrace : ∀ K : Set ℂ, IsCompact K → ∃ C : ℝ≥0∞, C < ∞ ∧ ∀ ε : ℝ, 0 < ε → |
| volume {x : ℝ | F (x+ε*Complex.I) ∈ K} ≤ C) : |
| ∀ᵐ ξ : ℂ, ∀ δ : ℝ, 0 < δ → |
| ∃ η : ℝ, 0 < η ∧ ∀ z : ℂ, 0 < z.im → |
| ‖F z-ξ‖ ≤ z.im^k/δ → η ≤ z.im := by |
| have hae : ∀ᵐ ξ : ℂ, ∀ R n : ℕ, |
| ξ ∉ F '' {z : ℂ | 0 < z.im} → ‖ξ‖ ≤ R → |
| ∃ η : ℝ, 0 < η ∧ ∀ z : ℂ, 0 < z.im → |
| ‖F z-ξ‖ ≤ z.im^k/(1/(n+1 : ℝ)) → η ≤ z.im := by |
| apply ae_all_iff.mpr |
| intro R |
| apply ae_all_iff.mpr |
| intro n |
| exact ae_omitted_positive_superlevel_height hk hL (by positivity) hcell htrace |
| filter_upwards [hae] with ξ hξ |
| intro δ hδ |
| by_cases ho : ξ ∈ F '' {z : ℂ | 0 < z.im} |
| · exact interior_positive_superlevel_height (by linarith) hδ hi hopen ho |
| obtain ⟨R,hR⟩ := exists_nat_ge ‖ξ‖ |
| obtain ⟨n,hn⟩ := exists_nat_gt (1/δ) |
| have hn0 : 0 < (n+1 : ℝ) := by positivity |
| have hnδ : 1/(n+1 : ℝ) ≤ δ := by |
| apply (div_le_iff₀ hn0).mpr |
| have hh := (div_lt_iff₀ hδ).mp hn |
| nlinarith |
| obtain ⟨η,hη,hηb⟩ := hξ R n ho hR |
| refine ⟨η,hη,fun z hz hzw => ?_⟩ |
| exact hηb z hz (hzw.trans (div_le_div_of_nonneg_left (Real.rpow_nonneg hz.le _) (by positivity) hnδ)) |
| |
| lemma ae_positive_superlevel_compact {F : ℂ → ℂ} {k L : ℝ} |
| (hk : 1 ≤ k) (hL : 0 ≤ L) |
| (hc : ContinuousOn F {z : ℂ | 0 < z.im}) |
| (hi : InjOn F {z : ℂ | 0 < z.im}) |
| (hopen : ∀ z : ℂ, 0 < z.im → map F (𝓝 z) = 𝓝 (F z)) |
| (ht : HasQuarticTail F (4*k)) |
| (hp : ∀ K : Set ℂ, IsCompact K → ∀ η : ℝ, 0 < η → |
| IsCompact ({z : ℂ | η ≤ z.im} ∩ F ⁻¹' K)) |
| (hcell : ∀ h x y x' : ℝ, ∀ ξ : ℂ, 0 < h → h ≤ y → y ≤ 2*h → |x'-x| ≤ h → |
| ξ ∉ F '' {z : ℂ | 0 < z.im} → |
| ‖F (x'+h*Complex.I)-ξ‖ ≤ L*‖F (x+y*Complex.I)-ξ‖) |
| (htrace : ∀ K : Set ℂ, IsCompact K → ∃ C : ℝ≥0∞, C < ∞ ∧ ∀ ε : ℝ, 0 < ε → |
| volume {x : ℝ | F (x+ε*Complex.I) ∈ K} ≤ C) : |
| ∀ᵐ ξ : ℂ, ∀ δ : ℝ, 0 < δ → IsCompact (closedPositiveSuperlevel F k δ ξ) := by |
| filter_upwards [ae_positive_superlevel_height hk hL hi hopen hcell htrace] with ξ hξ |
| intro δ hδ |
| exact positive_superlevel_compact (by linarith) hδ hc ht hp (hξ δ hδ) |
| |
| lemma class_ae_compact_superlevels (g : DiskClass) {β : ℝ} (hβ : 1 < β) |
| (hfin : (∫⁻ z in outsideCore, |
| ENNReal.ofReal (z.im^(β+1)*‖classFun g z‖⁻¹^4)) < ∞) : |
| ∀ᵐ ξ : ℂ, ∀ δ : ℝ, 0 < δ → |
| IsCompact (closedPositiveSuperlevel (classFun g) ((β+3)/4) δ ξ) := by |
| obtain ⟨M,hM,hMb⟩ := uniform_relative_class_bound |
| have ht : HasQuarticTail (classFun g) (4*((β+3)/4)) := by |
| convert class_quartic_tail g hβ hfin using 1; ring |
| apply ae_positive_superlevel_compact (by linarith : 1 ≤ (β+3)/4) |
| (by linarith : 0 ≤ 1+4*M) (classFun_schlicht g).1.1.continuousOn |
| (classFun_schlicht g).1.2 _ ht |
| (fun _ hK _ hη => class_proper_preimage g hβ hfin hK hη) |
| (fun _ _ _ _ _ hh hy hyy hxx hξ => class_omitted_cell_bound hM hMb g hh hy hyy hxx hξ) |
| (fun _ hK => class_trace_bound g hβ hfin hK) |
| intro z hz |
| have hd := ((classFun_schlicht g).1.1.analyticAt (isOpen_halfPlane.mem_nhds hz)).contDiffAt (n := 1) |
| exact (hd.hasStrictDerivAt (by norm_num)).map_nhds_eq |
| (univalent_deriv_ne_zero isOpen_halfPlane (classFun_schlicht g).1 hz) |
| |
| end Brennan |
| |
| end |
| |
| end OAI |
| |
