import OAI.Analysis.DiskMaximal.Basic namespace OAI open MeasureTheory Set open scoped ENNReal NNReal Topology ContDiff open scoped RealInnerProductSpace open MeasureTheory Set Filter open scoped Topology ENNReal open scoped Real RealInnerProductSpace noncomputable section namespace DiskGeometry variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] theorem mem_tangent_ball_iff {z y θ : E} (hθ : ‖θ‖ = 1) {L : ℝ} (hL : 0 < L) : y ∈ Metric.ball (z + L • θ) L ↔ ‖y - z‖ ^ 2 < 2 * L * ⟪y - z, θ⟫ := by rw [Metric.mem_ball, dist_eq_norm] have hv : y - (z + L • θ) = (y - z) - L • θ := by abel rw [hv, ← sq_lt_sq₀ (norm_nonneg _) hL.le, norm_sub_sq_real, norm_smul, Real.norm_eq_abs, abs_of_pos hL, hθ, mul_one, real_inner_smul_right] constructor <;> intro h <;> nlinarith theorem tangent_ball_mono {z θ : E} (hθ : ‖θ‖ = 1) {L L' : ℝ} (hL : 0 < L) (hLL' : L ≤ L') : Metric.ball (z + L • θ) L ⊆ Metric.ball (z + L' • θ) L' := by intro y hy rw [mem_tangent_ball_iff hθ hL] at hy rw [mem_tangent_ball_iff hθ (hL.trans_le hLL')] have hinner : 0 < ⟪y - z, θ⟫ := by by_contra hh have hh' : ⟪y - z, θ⟫ ≤ 0 := le_of_not_gt hh have := mul_nonpos_of_nonneg_of_nonpos (show 0 ≤ 2 * L by positivity) hh' nlinarith [sq_nonneg ‖y - z‖] exact hy.trans_le (mul_le_mul_of_nonneg_right (by linarith) hinner.le) def entryParameter (z y θ : E) : ℝ := ‖y - z‖ ^ 2 / (2 * ⟪y - z, θ⟫) theorem mem_tangent_ball_iff_entry {z y θ : E} (hθ : ‖θ‖ = 1) {L : ℝ} (hL : 0 < L) (ha : 0 < ⟪y - z, θ⟫) : y ∈ Metric.ball (z + L • θ) L ↔ entryParameter z y θ < L := by rw [mem_tangent_ball_iff hθ hL, entryParameter, div_lt_iff₀ (by positivity)] ring_nf /-- Uniform estimate on the affine difference of squared distances in a thin shell. -/ theorem shell_square_error {l δ d : ℝ} (hl : 0 ≤ l) (hl3 : l ≤ 3) (hδ : 0 ≤ δ) (hδ1 : δ ≤ 1 / 4) (hd : 0 ≤ d) (hclose : |d - l| ≤ 2 * δ) : |d ^ 2 - l ^ 2| ≤ 13 * δ := by have hd_upper : d ≤ l + 2 * δ := (abs_le.mp hclose).2 |> fun h => by linarith have hsum : 0 ≤ d + l := add_nonneg hd hl calc |d ^ 2 - l ^ 2| = |d - l| * (d + l) := by rw [show d ^ 2 - l ^ 2 = (d - l) * (d + l) by ring, abs_mul, abs_of_nonneg hsum] _ ≤ (2 * δ) * (d + l) := mul_le_mul_of_nonneg_right hclose hsum _ ≤ 13 * δ := by nlinarith /-- A shell point separated from the tangency point has positive radial projection. -/ theorem shell_projection_lower {ρ δ l a w2 : ℝ} (hρ : 0 < ρ) (hl : 0 < l) (hl3 : l ≤ 3) (hδ : δ ≤ ρ ^ 2 / 1600) (hw : ρ ^ 2 ≤ w2) (herr : |w2 - 2 * l * a| ≤ 13 * δ) : ρ ^ 2 / 12 ≤ a := by have hr2 : 0 < ρ ^ 2 := sq_pos_of_pos hρ have he := (abs_le.mp herr).2 have hla : 0 < a := by nlinarith nlinarith theorem entry_window {ρ δ l a w2 : ℝ} (hρ : 0 < ρ) (hδ : 0 ≤ δ) (ha : ρ ^ 2 / 12 ≤ a) (herr : |w2 - 2 * l * a| ≤ 13 * δ) : |w2 / (2 * a) - l| ≤ 78 * δ / ρ ^ 2 := by have hr2 : 0 < ρ ^ 2 := sq_pos_of_pos hρ have ha0 : 0 < a := lt_of_lt_of_le (by positivity) ha have h2a : 0 < 2 * a := by positivity rw [show w2 / (2 * a) - l = (w2 - 2 * l * a) / (2 * a) by field_simp, abs_div, abs_of_pos h2a] calc |w2 - 2 * l * a| / (2 * a) ≤ 13 * δ / (2 * a) := div_le_div_of_nonneg_right herr h2a.le _ ≤ 78 * δ / ρ ^ 2 := by apply (div_le_div_iff₀ h2a hr2).2 nlinarith [mul_le_mul_of_nonneg_right ha hδ] /-- Bound for the entry parameter for an actual shell near a tangent disk. -/ theorem shell_entry_window {z y θ : E} (hθ : ‖θ‖ = 1) {l ρ δ : ℝ} (hl : 1 / 2 ≤ l) (hl3 : l ≤ 3) (hρ : 0 < ρ) (hρ1 : ρ ≤ 1) (hδ : 0 < δ) (hδρ : δ ≤ ρ ^ 2 / 1600) (hsep : ρ ≤ ‖y - z‖) (hclose : |dist y (z + l • θ) - l| ≤ 2 * δ) : ρ ^ 2 / 12 ≤ ⟪y - z, θ⟫ ∧ |entryParameter z y θ - l| ≤ 78 * δ / ρ ^ 2 := by have hl0 : 0 < l := by linarith have hδ1 : δ ≤ 1 / 4 := by nlinarith have hsq : ρ ^ 2 ≤ ‖y - z‖ ^ 2 := (sq_le_sq₀ hρ.le (norm_nonneg _)).2 hsep have hd := shell_square_error hl0.le hl3 hδ.le hδ1 (dist_nonneg (x := y)) hclose have hv : y - (z + l • θ) = (y - z) - l • θ := by abel have herr : |‖y - z‖ ^ 2 - 2 * l * ⟪y - z, θ⟫| ≤ 13 * δ := by rw [dist_eq_norm, hv, norm_sub_sq_real, real_inner_smul_right, norm_smul, Real.norm_eq_abs, abs_of_pos hl0, hθ, mul_one] at hd convert hd using 1 congr 1 ring have ha := shell_projection_lower hρ hl0 hl3 hδρ hsq herr exact ⟨ha, entry_window hρ hδ.le ha herr⟩ theorem earlier_tangent_disk_subset {z θ : E} (hθ : ‖θ‖ = 1) {l ρ δ : ℝ} (hl : 1 / 2 ≤ l) (hl3 : l ≤ 3) (hρ : 0 < ρ) (hρ1 : ρ ≤ 1) (hδ : 0 < δ) (hδρ : δ ≤ ρ ^ 2 / 1600) : {y : E | ρ ≤ ‖y - z‖} ∩ Metric.ball (z + (l - 200 * δ / ρ ^ 2) • θ) (l - 200 * δ / ρ ^ 2) ⊆ Metric.ball (z + l • θ) (l - 2 * δ) := by let v : ℝ := 200 * δ / ρ ^ 2 have hr2 : 0 < ρ ^ 2 := sq_pos_of_pos hρ have hvpos : 0 < v := by dsimp [v]; positivity have hvsmall : v ≤ 1 / 8 := by dsimp [v] apply (div_le_iff₀ hr2).2 linarith have hv_eq : v * ρ ^ 2 = 200 * δ := by dsimp [v]; exact div_mul_cancel₀ _ hr2.ne' have hl0 : 0 < l := by linarith have hlv : 0 < l - v := by linarith have hδquarter : δ ≤ 1 / 4 := by nlinarith have hrad : 0 < l - 2 * δ := by nlinarith rintro y ⟨hsep, hy⟩ change y ∈ Metric.ball (z + (l - v) • θ) (l - v) at hy rw [mem_tangent_ball_iff hθ hlv] at hy have hnorm : ρ ^ 2 ≤ ‖y - z‖ ^ 2 := (sq_le_sq₀ hρ.le (norm_nonneg _)).2 hsep have hapos : 0 < ⟪y - z, θ⟫ := by by_contra ha have hmul := mul_nonpos_of_nonneg_of_nonpos (show 0 ≤ 2 * (l - v) by positivity) (le_of_not_gt ha) nlinarith have halower : ρ ^ 2 / 6 < ⟪y - z, θ⟫ := by nlinarith have hva := mul_lt_mul_of_pos_left halower hvpos have hsqdist : dist y (z + l • θ) ^ 2 < (l - 2 * δ) ^ 2 := by rw [dist_eq_norm, show y - (z + l • θ) = (y - z) - l • θ by abel, norm_sub_sq_real, real_inner_smul_right, norm_smul, Real.norm_eq_abs, abs_of_pos hl0, hθ, mul_one] nlinarith [mul_le_mul_of_nonneg_right hl3 hδ.le] exact (sq_lt_sq₀ dist_nonneg hrad.le).1 hsqdist end DiskGeometry namespace DiskGeometry /-- Cancellation of translated monotone integrals on a finite interval. -/ theorem monotone_increment_interval {G : ℝ → ℝ} (hG : Monotone G) {C v : ℝ} (h0 : ∀ x, 0 ≤ G x) (hC : ∀ x, G x ≤ C) (hv : 0 ≤ v) (a b : ℝ) : (∫ l in a..b, (G (l + v) - G (l - v))) ≤ 2 * v * C := by have hp : Monotone (fun l => G (l + v)) := hG.comp (by intro x y h; dsimp; linarith) have hm : Monotone (fun l => G (l - v)) := hG.comp (fun _ _ h => sub_le_sub_right h v) rw [intervalIntegral.integral_sub hp.intervalIntegrable hm.intervalIntegrable, intervalIntegral.integral_comp_add_right, intervalIntegral.integral_comp_sub_right, intervalIntegral.integral_interval_sub_interval_comm' hG.intervalIntegrable hG.intervalIntegrable hG.intervalIntegrable] have hn : 0 ≤ ∫ l in a - v..a + v, G l := intervalIntegral.integral_nonneg_of_forall (by linarith) h0 have hb : (∫ l in b - v..b + v, G l) ≤ 2 * v * C := by calc (∫ l in b - v..b + v, G l) ≤ ∫ l in b - v..b + v, C := intervalIntegral.integral_mono (by linarith) hG.intervalIntegrable intervalIntegrable_const hC _ = 2 * v * C := by rw [intervalIntegral.integral_const] change ((b + v) - (b - v)) * C = _ ring linarith theorem monotone_increment_integrable {G : ℝ → ℝ} (hG : Monotone G) {C v : ℝ} (h0 : ∀ x, 0 ≤ G x) (hC : ∀ x, G x ≤ C) (hv : 0 ≤ v) : Integrable (fun l => G (l + v) - G (l - v)) ∧ (∫ l, (G (l + v) - G (l - v))) ≤ 2 * v * C := by have hp : Monotone (fun l => G (l + v)) := hG.comp (by intro x y h; dsimp; linarith) have hm : Monotone (fun l => G (l - v)) := hG.comp (fun _ _ h => sub_le_sub_right h v) have hn (l : ℝ) : 0 ≤ G (l + v) - G (l - v) := sub_nonneg.mpr (hG (by linarith)) have hi (a b : ℝ) : IntervalIntegrable (fun l => G (l + v) - G (l - v)) volume a b := hp.intervalIntegrable.sub hm.intervalIntegrable have htop : Tendsto (fun n : ℕ => (n : ℝ)) atTop atTop := tendsto_natCast_atTop_atTop have hbot : Tendsto (fun n : ℕ => -(n : ℝ)) atTop atBot := tendsto_neg_atTop_atBot.comp htop have hint : Integrable (fun l => G (l + v) - G (l - v)) := by apply integrable_of_intervalIntegral_norm_bounded (2 * v * C) (fun n : ℕ => (hi (-(n : ℝ)) n).1) hbot htop exact Eventually.of_forall (fun n => by simp_rw [Real.norm_of_nonneg (hn _)] exact monotone_increment_interval hG h0 hC hv _ _) refine ⟨hint, ?_⟩ exact le_of_tendsto (intervalIntegral_tendsto_integral hint hbot htop) (Eventually.of_forall (fun n => monotone_increment_interval hG h0 hC hv (-(n : ℝ)) n)) /-- The permitted jump times have uniformly bounded length, independent of the size of the original positive measure. Only its truncated distribution is used. -/ theorem monotone_jump_window_measure {H : ℝ → ℝ} (hH : Monotone H) (hH0 : ∀ l, 0 ≤ H l) {T U v : ℝ} (hT : 0 ≤ T) (hU : 0 < U) (hv : 0 ≤ v) : volume {l | H (l - v) ≤ T ∧ U ≤ H (l + v) - H (l - v)} ≤ ENNReal.ofReal (2 * v * (1 + T / U)) := by let G : ℝ → ℝ := fun l => min (H l) (T + U) have hG : Monotone G := hH.min monotone_const have hG0 : ∀ l, 0 ≤ G l := fun l => le_min (hH0 l) (by positivity) have hGC : ∀ l, G l ≤ T + U := fun l => min_le_right _ _ obtain ⟨hi, hb⟩ := monotone_increment_integrable hG hG0 hGC hv have hn (l : ℝ) : 0 ≤ G (l + v) - G (l - v) := sub_nonneg.mpr (hG (by linarith)) have hmi : Integrable (fun l => (G (l + v) - G (l - v)) / U) := hi.div_const U have hm := hmi.measure_le_integral (Eventually.of_forall (fun l => div_nonneg (hn l) hU.le)) (s := {l | H (l - v) ≤ T ∧ U ≤ H (l + v) - H (l - v)}) (by intro l hl rw [le_div_iff₀ hU, one_mul] have hd : G (l - v) = H (l - v) := min_eq_left (by linarith [hl.1]) rw [hd] change U ≤ min (H (l + v)) (T + U) - H (l - v) rw [le_sub_iff_add_le, le_min_iff] exact ⟨by linarith [hl.2], by linarith [hl.1]⟩) calc volume {l | H (l - v) ≤ T ∧ U ≤ H (l + v) - H (l - v)} ≤ ENNReal.ofReal (∫ l, (G (l + v) - G (l - v)) / U) := hm _ ≤ ENNReal.ofReal (2 * v * (1 + T / U)) := by apply ENNReal.ofReal_le_ofReal rw [integral_div] calc (∫ l, G (l + v) - G (l - v)) / U ≤ (2 * v * (T + U)) / U := div_le_div_of_nonneg_right hb hU.le _ = 2 * v * (1 + T / U) := by field_simp; ring end DiskGeometry namespace DiskGeometry /-- A polar-coordinate area bound from uniform lengths of radial sections. -/ theorem volume_le_radial_sections {s : Set ℂ} (hs : MeasurableSet s) (z : ℂ) {R : ℝ} (hR : 0 ≤ R) {B : ℝ≥0∞} (hbound : ∀ x ∈ s, ‖x - z‖ ≤ R) (hsections : ∀ θ : ℝ, volume {r : ℝ | 0 < r ∧ z + (r : ℂ) * ((Real.cos θ : ℂ) + (Real.sin θ : ℂ) * Complex.I) ∈ s} ≤ B) : volume s ≤ ENNReal.ofReal (2 * Real.pi * R) * B := by classical let f : ℂ → ℝ≥0∞ := s.indicator 1 have hf : Measurable f := measurable_const.indicator hs have hfc : Measurable (fun p : ℝ × ℝ => f (z + Complex.polarCoord.symm p)) := by apply hf.comp simp only [Complex.polarCoord_symm_apply] fun_prop have hpolar : volume s = ∫⁻ p in polarCoord.target, ENNReal.ofReal p.1 * f (z + Complex.polarCoord.symm p) := by rw [← lintegral_indicator_one hs] change (∫⁻ x, f x) = _ rw [← lintegral_add_left_eq_self f z, ← Complex.lintegral_comp_polarCoord_symm (fun x => f (z + x))] rfl have hinner (θ : ℝ) : (∫⁻ r in Ioi (0 : ℝ), f (z + Complex.polarCoord.symm (r, θ))) ≤ B := by let A : Set ℝ := {r | 0 < r ∧ z + (r : ℂ) * ((Real.cos θ : ℂ) + (Real.sin θ : ℂ) * Complex.I) ∈ s} have hA : MeasurableSet A := measurableSet_Ioi.inter (hs.preimage (by fun_prop)) calc _ = ∫⁻ r, A.indicator 1 r := by rw [← lintegral_indicator measurableSet_Ioi] apply lintegral_congr intro r by_cases hr : 0 < r <;> by_cases hx : z + (r : ℂ) * ((Real.cos θ : ℂ) + (Real.sin θ : ℂ) * Complex.I) ∈ s <;> simp only [f, A, Complex.polarCoord_symm_apply, Set.indicator_apply, mem_Ioi, mem_ofPred_eq, Pi.one_apply, hr, hx, and_true, and_false, ite_true, ite_false] _ = volume A := lintegral_indicator_one hA _ ≤ B := hsections θ calc volume s = ∫⁻ p in polarCoord.target, ENNReal.ofReal p.1 * f (z + Complex.polarCoord.symm p) := hpolar _ ≤ ∫⁻ p in polarCoord.target, ENNReal.ofReal R * f (z + Complex.polarCoord.symm p) := by apply setLIntegral_mono' polarCoord.open_target.measurableSet intro p hp by_cases hx : z + Complex.polarCoord.symm p ∈ s · gcongr have hn := hbound _ hx rw [add_sub_cancel_left, Complex.norm_polarCoord_symm, abs_of_pos hp.1] at hn exact hn · simp only [f, indicator_of_notMem hx, mul_zero, le_refl] _ = ENNReal.ofReal R * ∫⁻ θ in Ioo (-Real.pi) Real.pi, ∫⁻ r in Ioi (0 : ℝ), f (z + Complex.polarCoord.symm (r, θ)) := by rw [lintegral_const_mul _ hfc] congr 1 exact setLIntegral_prod_symm _ hfc.aemeasurable _ ≤ ENNReal.ofReal R * ∫⁻ θ in Ioo (-Real.pi) Real.pi, B := by gcongr with θ exact hinner θ _ = ENNReal.ofReal (2 * Real.pi * R) * B := by rw [lintegral_const, Measure.restrict_apply_univ, Real.volume_Ioo] rw [show Real.pi - -Real.pi = 2 * Real.pi by ring] rw [mul_comm (2 * Real.pi) R, ENNReal.ofReal_mul hR] ac_rfl end DiskGeometry namespace DiskGeometry /-- Mass in the increasing family of tangent disks, away from its tangency cap. -/ def tangentMass (S : Measure ℂ) (ρ : ℝ) (z θ : ℂ) (L : ℝ) : ℝ := (S ({y : ℂ | ρ ≤ ‖y - z‖} ∩ Metric.ball (z + L • θ) L)).toReal theorem tangentMass_nonneg (S : Measure ℂ) (ρ : ℝ) (z θ : ℂ) (L : ℝ) : 0 ≤ tangentMass S ρ z θ L := ENNReal.toReal_nonneg theorem tangentMass_mono (S : Measure ℂ) [IsFiniteMeasure S] (ρ : ℝ) (z θ : ℂ) (hθ : ‖θ‖ = 1) : Monotone (tangentMass S ρ z θ) := by intro L L' hLL' by_cases hL : 0 < L · apply ENNReal.toReal_mono (measure_ne_top S _) exact measure_mono (inter_subset_inter_right _ (tangent_ball_mono hθ hL hLL')) · have hball : Metric.ball (z + L • θ) L = ∅ := Metric.ball_eq_empty.mpr (le_of_not_gt hL) simp only [tangentMass, hball, inter_empty, measure_empty, ENNReal.toReal_zero] exact ENNReal.toReal_nonneg theorem measurable_tangentMass (S : Measure ℂ) [IsFiniteMeasure S] (ρ : ℝ) {α : Type*} [MeasurableSpace α] {z θ : α → ℂ} {L : α → ℝ} (hz : Measurable z) (hθ : Measurable θ) (hL : Measurable L) : Measurable (fun a => tangentMass S ρ (z a) (θ a) (L a)) := by have hm : MeasurableSet {p : α × ℂ | ρ ≤ ‖p.2 - z p.1‖ ∧ dist p.2 (z p.1 + L p.1 • θ p.1) < L p.1} := by exact (measurableSet_le measurable_const ((measurable_snd.sub (hz.comp measurable_fst)).norm)).inter (measurableSet_lt (measurable_snd.dist ((hz.comp measurable_fst).add ((hL.comp measurable_fst).smul (hθ.comp measurable_fst)))) (hL.comp measurable_fst)) exact (measurable_measure_prodMk_left (ν := S) hm).ennreal_toReal /-- The measurable condition whose planar cost is bounded by the ray-length estimate. -/ def rayCondition (S : Measure ℂ) (ρ T U v : ℝ) (x z : ℂ) : Prop := 1 / 2 ≤ ‖x - z‖ ∧ ‖x - z‖ ≤ 3 ∧ tangentMass S ρ z (‖x - z‖⁻¹ • (x - z)) (‖x - z‖ - v) ≤ T ∧ U ≤ tangentMass S ρ z (‖x - z‖⁻¹ • (x - z)) (‖x - z‖ + v) - tangentMass S ρ z (‖x - z‖⁻¹ • (x - z)) (‖x - z‖ - v) theorem measurableSet_rayCondition (S : Measure ℂ) [IsFiniteMeasure S] (ρ T U v : ℝ) : MeasurableSet {p : ℂ × ℂ | rayCondition S ρ T U v p.1 p.2} := by have hn : Measurable (fun p : ℂ × ℂ => ‖p.1 - p.2‖) := by fun_prop have hdir : Measurable (fun p : ℂ × ℂ => ‖p.1 - p.2‖⁻¹ • (p.1 - p.2)) := by fun_prop have hm := measurable_tangentMass S ρ measurable_snd hdir (hn.sub_const v) have hp := measurable_tangentMass S ρ measurable_snd hdir (hn.add_const v) exact (measurableSet_le measurable_const hn).inter ((measurableSet_le hn measurable_const).inter ((measurableSet_le hm measurable_const).inter (measurableSet_le measurable_const (hp.sub hm)))) /-- Full two-dimensional ray-condition estimate, independent of the chosen radius. -/ theorem volume_rayCondition (S : Measure ℂ) [IsFiniteMeasure S] (ρ : ℝ) {T U v : ℝ} (hT : 0 ≤ T) (hU : 0 < U) (hv : 0 ≤ v) (z : ℂ) : volume {x : ℂ | rayCondition S ρ T U v x z} ≤ ENNReal.ofReal (6 * Real.pi) * ENNReal.ofReal (2 * v * (1 + T / U)) := by have hs : MeasurableSet {x : ℂ | rayCondition S ρ T U v x z} := (measurableSet_rayCondition S ρ T U v).preimage (show Measurable (fun x : ℂ => (x, z)) by fun_prop) have hbound : ∀ x ∈ {x : ℂ | rayCondition S ρ T U v x z}, ‖x - z‖ ≤ 3 := fun _ h => h.2.1 have hsections (a : ℝ) : volume {r : ℝ | 0 < r ∧ z + (r : ℂ) * ((Real.cos a : ℂ) + (Real.sin a : ℂ) * Complex.I) ∈ {x : ℂ | rayCondition S ρ T U v x z}} ≤ ENNReal.ofReal (2 * v * (1 + T / U)) := by let θ : ℂ := (Real.cos a : ℂ) + (Real.sin a : ℂ) * Complex.I have hθ : ‖θ‖ = 1 := by simpa only [θ, Complex.ofReal_cos, Complex.ofReal_sin] using Complex.norm_cos_add_sin_mul_I a apply le_trans (measure_mono (t := {l : ℝ | tangentMass S ρ z θ (l - v) ≤ T ∧ U ≤ tangentMass S ρ z θ (l + v) - tangentMass S ρ z θ (l - v)}) ?_) (monotone_jump_window_measure (tangentMass_mono S ρ z θ hθ) (tangentMass_nonneg S ρ z θ) hT hU hv) rintro r ⟨hr, hx⟩ have hnorm : ‖z + (r : ℂ) * θ - z‖ = r := by rw [add_sub_cancel_left, norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_pos hr, hθ, mul_one] have hdir : ‖z + (r : ℂ) * θ - z‖⁻¹ • (z + (r : ℂ) * θ - z) = θ := by rw [hnorm, add_sub_cancel_left, ← Complex.real_smul, smul_smul, inv_mul_cancel₀ hr.ne', one_smul] change rayCondition S ρ T U v (z + (r : ℂ) * θ) z at hx rw [rayCondition] at hx rw [hdir, hnorm] at hx exact hx.2.2 simpa only [show (2 : ℝ) * Real.pi * 3 = 6 * Real.pi by ring] using volume_le_radial_sections hs z (by norm_num : (0 : ℝ) ≤ 3) hbound hsections end DiskGeometry namespace DiskGeometry def thinShell (x : ℂ) (r δ : ℝ) : Set ℂ := {y | |‖y - x‖ - r| ≤ δ} theorem measurableSet_thinShell (x : ℂ) (r δ : ℝ) : MeasurableSet (thinShell x r δ) := by exact measurableSet_le (by fun_prop) measurable_const theorem shell_center_bounds {x z : ℂ} {r δ : ℝ} (hr : r ∈ Icc (1 : ℝ) 2) (hδ : δ ≤ 1 / 4) (hz : z ∈ thinShell x r δ) : 1 / 2 ≤ ‖x - z‖ ∧ ‖x - z‖ ≤ 3 := by change |‖z - x‖ - r| ≤ δ at hz have hh := abs_le.mp hz rw [norm_sub_rev z x] at hh constructor <;> linarith [hr.1, hr.2] theorem shell_pair_close {x y z : ℂ} {r δ : ℝ} (hy : y ∈ thinShell x r δ) (hz : z ∈ thinShell x r δ) : |‖y - x‖ - ‖x - z‖| ≤ 2 * δ := by change |‖y - x‖ - r| ≤ δ at hy change |‖z - x‖ - r| ≤ δ at hz have hy' := abs_le.mp hy have hz' := abs_le.mp hz rw [norm_sub_rev z x] at hz' exact abs_le.mpr ⟨by linarith [hy'.1, hz'.2], by linarith [hy'.2, hz'.1]⟩ /-- Every distant point of the thin shell crosses the tangent disks in the short window. -/ theorem shell_subset_tangent_difference {z θ : ℂ} (hθ : ‖θ‖ = 1) {l ρ δ : ℝ} (hl : 1 / 2 ≤ l) (hl3 : l ≤ 3) (hρ : 0 < ρ) (hρ1 : ρ ≤ 1) (hδ : 0 < δ) (hδρ : δ ≤ ρ ^ 2 / 1600) : {y : ℂ | ρ ≤ ‖y - z‖ ∧ |‖y - (z + l • θ)‖ - l| ≤ 2 * δ} ⊆ ({y : ℂ | ρ ≤ ‖y - z‖} ∩ Metric.ball (z + (l + 200 * δ / ρ ^ 2) • θ) (l + 200 * δ / ρ ^ 2)) \ ({y : ℂ | ρ ≤ ‖y - z‖} ∩ Metric.ball (z + (l - 200 * δ / ρ ^ 2) • θ) (l - 200 * δ / ρ ^ 2)) := by let v : ℝ := 200 * δ / ρ ^ 2 have hr2 : 0 < ρ ^ 2 := sq_pos_of_pos hρ have hvp : 0 < v := by dsimp [v]; positivity have hvs : v ≤ 1 / 8 := by dsimp [v] apply (div_le_iff₀ hr2).2 linarith have hlm : 0 < l - v := by linarith have hlp : 0 < l + v := by linarith have h78 : 78 * δ / ρ ^ 2 < v := by dsimp [v] exact (div_lt_div_iff_of_pos_right hr2).2 (by linarith) rintro y ⟨hsep, hclose⟩ obtain ⟨ha, hw⟩ := shell_entry_window hθ hl hl3 hρ hρ1 hδ hδρ hsep (by simpa only [dist_eq_norm] using hclose) have ha0 : 0 < ⟪y - z, θ⟫ := lt_of_lt_of_le (by positivity) ha have hentrylo : l - v < entryParameter z y θ := by linarith [(abs_le.mp hw).1] have hentryhi : entryParameter z y θ < l + v := by linarith [(abs_le.mp hw).2] refine ⟨⟨hsep, (mem_tangent_ball_iff_entry hθ hlp ha0).2 hentryhi⟩, ?_⟩ rintro ⟨_, hm⟩ have hm' := (mem_tangent_ball_iff_entry hθ hlm ha0).1 hm linarith /-- The shell mass away from a cap is bounded by the one-dimensional tangent-mass jump. -/ theorem shell_mass_le_jump (S : Measure ℂ) [IsFiniteMeasure S] {x z : ℂ} {r ρ δ : ℝ} (hr : r ∈ Icc (1 : ℝ) 2) (hρ : 0 < ρ) (hρ1 : ρ ≤ 1) (hδ : 0 < δ) (hδρ : δ ≤ ρ ^ 2 / 1600) (hz : z ∈ thinShell x r δ) : (S (thinShell x r δ \ Metric.ball z ρ)).toReal ≤ tangentMass S ρ z (‖x - z‖⁻¹ • (x - z)) (‖x - z‖ + 200 * δ / ρ ^ 2) - tangentMass S ρ z (‖x - z‖⁻¹ • (x - z)) (‖x - z‖ - 200 * δ / ρ ^ 2) := by let l : ℝ := ‖x - z‖ let θ : ℂ := l⁻¹ • (x - z) have hδquarter : δ ≤ 1 / 4 := by nlinarith obtain ⟨hl, hl3⟩ := shell_center_bounds hr hδquarter hz have hl0 : 0 < l := by dsimp [l]; linarith have hθ : ‖θ‖ = 1 := by dsimp only [θ] rw [norm_smul, Real.norm_of_nonneg (inv_nonneg.mpr hl0.le)] exact inv_mul_cancel₀ hl0.ne' have hx : z + l • θ = x := by dsimp only [θ] rw [smul_smul, mul_inv_cancel₀ hl0.ne', one_smul] abel let A := {y : ℂ | ρ ≤ ‖y - z‖} ∩ Metric.ball (z + (l - 200 * δ / ρ ^ 2) • θ) (l - 200 * δ / ρ ^ 2) let B := {y : ℂ | ρ ≤ ‖y - z‖} ∩ Metric.ball (z + (l + 200 * δ / ρ ^ 2) • θ) (l + 200 * δ / ρ ^ 2) have hAm : MeasurableSet A := (measurableSet_le measurable_const (by fun_prop)).inter Metric.isOpen_ball.measurableSet have hsub : thinShell x r δ \ Metric.ball z ρ ⊆ B \ A := by apply subset_trans _ (shell_subset_tangent_difference hθ hl hl3 hρ hρ1 hδ hδρ) rintro y ⟨hy, hyz⟩ exact ⟨le_of_not_gt (by simpa only [Metric.mem_ball, dist_eq_norm] using hyz), by rw [hx]; exact shell_pair_close hy hz⟩ have hAB : A ⊆ B := by apply inter_subset_inter_right apply tangent_ball_mono hθ · have : 200 * δ / ρ ^ 2 ≤ 1 / 8 := by apply (div_le_iff₀ (sq_pos_of_pos hρ)).2 linarith change 0 < l - 200 * δ / ρ ^ 2 linarith · have : 0 ≤ 200 * δ / ρ ^ 2 := by positivity linarith have hmass := ENNReal.toReal_mono (measure_ne_top S (B \ A)) (measure_mono hsub) rw [measure_sdiff hAB hAm.nullMeasurableSet (measure_ne_top S A), ENNReal.toReal_sub_of_le (measure_mono hAB) (measure_ne_top S B)] at hmass exact hmass /-- The earlier tangent-disk mass is controlled by the concentric interior mass. -/ theorem earlier_mass_le_interior (S : Measure ℂ) [IsFiniteMeasure S] {x z : ℂ} {r ρ δ : ℝ} (hr : r ∈ Icc (1 : ℝ) 2) (hρ : 0 < ρ) (hρ1 : ρ ≤ 1) (hδ : 0 < δ) (hδρ : δ ≤ ρ ^ 2 / 1600) (hz : z ∈ thinShell x r δ) : tangentMass S ρ z (‖x - z‖⁻¹ • (x - z)) (‖x - z‖ - 200 * δ / ρ ^ 2) ≤ (S (Metric.ball x (r - δ))).toReal := by let l : ℝ := ‖x - z‖ let θ : ℂ := l⁻¹ • (x - z) have hδquarter : δ ≤ 1 / 4 := by nlinarith obtain ⟨hl, hl3⟩ := shell_center_bounds hr hδquarter hz have hl0 : 0 < l := by dsimp [l]; linarith have hθ : ‖θ‖ = 1 := by dsimp only [θ] rw [norm_smul, Real.norm_of_nonneg (inv_nonneg.mpr hl0.le)] exact inv_mul_cancel₀ hl0.ne' have hx : z + l • θ = x := by dsimp only [θ] rw [smul_smul, mul_inv_cancel₀ hl0.ne', one_smul] abel apply ENNReal.toReal_mono (measure_ne_top S _) apply measure_mono apply (earlier_tangent_disk_subset hθ hl hl3 hρ hρ1 hδ hδρ).trans rw [hx] apply Metric.ball_subset_ball change |‖z - x‖ - r| ≤ δ at hz have hh := (abs_le.mp hz).2 rw [norm_sub_rev z x] at hh change l - 2 * δ ≤ r - δ dsimp [l] linarith end DiskGeometry namespace DiskGeometry /-- Area of a strip with continuously moving interval fibers. -/ theorem volume_interval_fibers (c : ℝ → ℝ) (hc : Continuous c) (lo hi w : ℝ) : volume {p : ℝ × ℝ | p.2 ∈ Icc lo hi ∧ p.1 ∈ Icc (c p.2 - w) (c p.2 + w)} = ENNReal.ofReal (hi - lo) * ENNReal.ofReal (2 * w) := by have hs : MeasurableSet {p : ℝ × ℝ | p.2 ∈ Icc lo hi ∧ p.1 ∈ Icc (c p.2 - w) (c p.2 + w)} := by exact (measurableSet_Icc.preimage continuous_snd.measurable).inter ((isClosed_le (hc.comp continuous_snd |>.sub continuous_const) continuous_fst).measurableSet.inter (isClosed_le continuous_fst (hc.comp continuous_snd |>.add continuous_const)).measurableSet) change (volume.prod volume) _ = _ rw [Measure.prod_apply_symm hs] have hf (y : ℝ) : volume ((fun x : ℝ => (x, y)) ⁻¹' {p : ℝ × ℝ | p.2 ∈ Icc lo hi ∧ p.1 ∈ Icc (c p.2 - w) (c p.2 + w)}) = (Icc lo hi).indicator (fun _ => ENNReal.ofReal (2 * w)) y := by by_cases hy : y ∈ Icc lo hi · simp only [preimage_ofPred_eq, hy, true_and, indicator_of_mem hy] change volume (Icc (c y - w) (c y + w)) = _ rw [Real.volume_Icc] congr 1 ring · simp only [preimage_ofPred_eq, hy, false_and, ofPred_false, measure_empty, indicator_of_notMem hy] simp_rw [hf] rw [lintegral_indicator measurableSet_Icc, lintegral_const, Measure.restrict_apply_univ, Real.volume_Icc, mul_comm] /-- A nonvertical affine strip cut off in the second coordinate. -/ theorem volume_affine_strip {a : ℝ} (ha : a ≠ 0) (b c d R w : ℝ) : volume {p : ℝ × ℝ | |p.2 - d| ≤ R ∧ |a * p.1 + b * p.2 + c| ≤ w} = ENNReal.ofReal (2 * R) * ENNReal.ofReal (2 * (w / |a|)) := by have hs : {p : ℝ × ℝ | |p.2 - d| ≤ R ∧ |a * p.1 + b * p.2 + c| ≤ w} = {p : ℝ × ℝ | p.2 ∈ Icc (d - R) (d + R) ∧ p.1 ∈ Icc (-(b * p.2 + c) / a - w / |a|) (-(b * p.2 + c) / a + w / |a|)} := by ext p simp only [mem_ofPred_eq, mem_Icc] have heq : a * p.1 + b * p.2 + c = a * (p.1 - -(b * p.2 + c) / a) := by field_simp ring rw [heq, abs_mul, ← le_div_iff₀' (abs_pos.mpr ha)] simp only [abs_le] constructor <;> rintro ⟨⟨h1, h2⟩, h3, h4⟩ <;> constructor <;> constructor <;> linarith rw [hs] rw [volume_interval_fibers (fun y => -(b * y + c) / a) (by fun_prop)] congr 2 ring /-- At least one coordinate sees half the Euclidean separation. -/ theorem complex_coordinate_separation {z : ℂ} {ρ : ℝ} (hz : ρ ≤ ‖z‖) : ρ / 2 ≤ |z.re| ∨ ρ / 2 ≤ |z.im| := by by_contra h push Not at h have hnorm : ‖z‖ ≤ |z.re| + |z.im| := Complex.norm_le_abs_re_add_abs_im z linarith /-- The centers of two separated points lying on an almost common circle form an affine strip of small area. No selected radius occurs here. -/ theorem volume_two_point_strip {y z : ℂ} {ρ δ : ℝ} (hρ : 0 < ρ) (hδ : 0 ≤ δ) (hyz : ρ ≤ ‖y - z‖) : volume {x : ℂ | ‖x - z‖ ≤ 3 ∧ |‖x - y‖ ^ 2 - ‖x - z‖ ^ 2| ≤ 13 * δ} ≤ ENNReal.ofReal (156 * δ / ρ) := by let a : ℝ := 2 * (z.re - y.re) let b : ℝ := 2 * (z.im - y.im) let c : ℝ := ‖y‖ ^ 2 - ‖z‖ ^ 2 have halg (x : ℂ) : ‖x - y‖ ^ 2 - ‖x - z‖ ^ 2 = a * x.re + b * x.im + c := by simp only [Complex.sq_norm, Complex.normSq_apply, Complex.sub_re, Complex.sub_im, a, b, c] ring have hsplit := complex_coordinate_separation hyz have hmain (a b c d : ℝ) (ha : ρ ≤ |a|) : volume {p : ℝ × ℝ | |p.2 - d| ≤ 3 ∧ |a * p.1 + b * p.2 + c| ≤ 13 * δ} ≤ ENNReal.ofReal (156 * δ / ρ) := by have hap : 0 < |a| := hρ.trans_le ha rw [volume_affine_strip (abs_pos.mp hap)] rw [← ENNReal.ofReal_mul (by norm_num)] apply ENNReal.ofReal_le_ofReal calc (2 * 3) * (2 * (13 * δ / |a|)) = 156 * δ / |a| := by ring _ ≤ 156 * δ / ρ := div_le_div_of_nonneg_left (by positivity) hρ ha rcases hsplit with hre | him · have ha : ρ ≤ |a| := by dsimp [a] rw [abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 2), abs_sub_comm] simpa only [Complex.sub_re] using (show ρ ≤ 2 * |(y - z).re| by linarith) let s : Set (ℝ × ℝ) := {p | |p.2 - z.im| ≤ 3 ∧ |a * p.1 + b * p.2 + c| ≤ 13 * δ} have hsub : {x : ℂ | ‖x - z‖ ≤ 3 ∧ |‖x - y‖ ^ 2 - ‖x - z‖ ^ 2| ≤ 13 * δ} ⊆ Complex.measurableEquivRealProd ⁻¹' s := by intro x hx exact ⟨(Complex.abs_im_le_norm (x - z)).trans hx.1, by simpa [halg x] using hx.2⟩ have hs : MeasurableSet s := by exact (isClosed_le (show Continuous (fun p : ℝ × ℝ => |p.2 - z.im|) from by fun_prop) continuous_const).measurableSet.inter (isClosed_le (show Continuous (fun p : ℝ × ℝ => |a * p.1 + b * p.2 + c|) from by fun_prop) continuous_const).measurableSet calc _ ≤ volume (Complex.measurableEquivRealProd ⁻¹' s) := measure_mono hsub _ = volume s := Complex.volume_preserving_equiv_real_prod.measure_preimage hs.nullMeasurableSet _ ≤ _ := hmain a b c z.im ha · have hb : ρ ≤ |b| := by dsimp [b] rw [abs_mul, abs_of_pos (by norm_num : (0 : ℝ) < 2), abs_sub_comm] simpa only [Complex.sub_im] using (show ρ ≤ 2 * |(y - z).im| by linarith) let s : Set (ℝ × ℝ) := {p | |p.2 - z.re| ≤ 3 ∧ |b * p.1 + a * p.2 + c| ≤ 13 * δ} have hmp : MeasurePreserving (fun x : ℂ => (x.im, x.re)) := Measure.measurePreserving_swap.comp Complex.volume_preserving_equiv_real_prod have hsub : {x : ℂ | ‖x - z‖ ≤ 3 ∧ |‖x - y‖ ^ 2 - ‖x - z‖ ^ 2| ≤ 13 * δ} ⊆ (fun x : ℂ => (x.im, x.re)) ⁻¹' s := by intro x hx refine ⟨(Complex.abs_re_le_norm (x - z)).trans hx.1, ?_⟩ simpa only [halg x, add_comm (a * x.re) (b * x.im)] using hx.2 have hs : MeasurableSet s := by exact (isClosed_le (show Continuous (fun p : ℝ × ℝ => |p.2 - z.re|) from by fun_prop) continuous_const).measurableSet.inter (isClosed_le (show Continuous (fun p : ℝ × ℝ => |b * p.1 + a * p.2 + c|) from by fun_prop) continuous_const).measurableSet calc _ ≤ volume ((fun x : ℂ => (x.im, x.re)) ⁻¹' s) := measure_mono hsub _ = volume s := hmp.measure_preimage hs.nullMeasurableSet _ ≤ _ := hmain b a c z.re hb end DiskGeometry namespace DiskGeometry def pairCondition (ρ δ : ℝ) (x z y : ℂ) : Prop := 1 / 2 ≤ ‖x - z‖ ∧ ‖x - z‖ ≤ 3 ∧ |‖y - x‖ - ‖x - z‖| ≤ 2 * δ ∧ ρ ≤ ‖y - z‖ theorem measurableSet_pairCondition (ρ δ : ℝ) : MeasurableSet {p : (ℂ × ℂ) × ℂ | pairCondition ρ δ p.1.1 p.1.2 p.2} := by have hn : Measurable (fun p : (ℂ × ℂ) × ℂ => ‖p.1.1 - p.1.2‖) := by fun_prop have hc : Measurable (fun p : (ℂ × ℂ) × ℂ => |‖p.2 - p.1.1‖ - ‖p.1.1 - p.1.2‖|) := by fun_prop have hs : Measurable (fun p : (ℂ × ℂ) × ℂ => ‖p.2 - p.1.2‖) := by fun_prop exact (measurableSet_le measurable_const hn).inter ((measurableSet_le hn measurable_const).inter ((measurableSet_le hc measurable_const).inter (measurableSet_le measurable_const hs))) theorem volume_pairCondition {ρ δ : ℝ} (hρ : 0 < ρ) (hδ : 0 ≤ δ) (hδquarter : δ ≤ 1 / 4) (z y : ℂ) : volume {x : ℂ | pairCondition ρ δ x z y} ≤ ENNReal.ofReal (156 * δ / ρ) := by by_cases hsep : ρ ≤ ‖y - z‖ · apply le_trans (measure_mono (t := {x : ℂ | ‖x - z‖ ≤ 3 ∧ |‖x - y‖ ^ 2 - ‖x - z‖ ^ 2| ≤ 13 * δ}) ?_) (volume_two_point_strip hρ hδ hsep) intro x hx obtain ⟨hl, hl3, hc, _⟩ := hx refine ⟨hl3, ?_⟩ rw [norm_sub_rev x y] exact shell_square_error (norm_nonneg _) hl3 hδ hδquarter (norm_nonneg _) hc · have he : {x : ℂ | pairCondition ρ δ x z y} = ∅ := by ext x simp only [mem_ofPred_eq, mem_empty_iff_false, iff_false] exact fun h => hsep h.2.2.2 rw [he, measure_empty] exact bot_le def pairMass (ν : Measure ℂ) (ρ δ : ℝ) (x z : ℂ) : ℝ≥0∞ := ν {y : ℂ | pairCondition ρ δ x z y} theorem measurable_pairMass (ν : Measure ℂ) [IsFiniteMeasure ν] (ρ δ : ℝ) : Measurable (fun p : ℂ × ℂ => pairMass ν ρ δ p.1 p.2) := measurable_measure_prodMk_left (measurableSet_pairCondition ρ δ) def rayIndicator (S : Measure ℂ) (ρ T U v : ℝ) (x z : ℂ) : ℝ≥0∞ := {p : ℂ × ℂ | rayCondition S ρ T U v p.1 p.2}.indicator 1 (x, z) theorem measurable_rayIndicator (S : Measure ℂ) [IsFiniteMeasure S] (ρ T U v : ℝ) : Measurable (fun p : ℂ × ℂ => rayIndicator S ρ T U v p.1 p.2) := measurable_const.indicator (measurableSet_rayCondition S ρ T U v) /-- The first contribution to the Borel majorant is controlled by the two-point strip. -/ theorem lintegral_pairMass (ν : Measure ℂ) [IsFiniteMeasure ν] {ρ δ : ℝ} (hρ : 0 < ρ) (hδ : 0 ≤ δ) (hδquarter : δ ≤ 1 / 4) : (∫⁻ x : ℂ, ∫⁻ z : ℂ, pairMass ν ρ δ x z ∂ν) ≤ ENNReal.ofReal (156 * δ / ρ) * ν univ * ν univ := by rw [lintegral_lintegral_swap (measurable_pairMass ν ρ δ).aemeasurable] apply le_trans (lintegral_mono (fun z => ?_)) (by rw [lintegral_const]) let A : Set (ℂ × ℂ) := {p | pairCondition ρ δ p.1 z p.2} have hA : MeasurableSet A := (measurableSet_pairCondition ρ δ).preimage (show Measurable (fun p : ℂ × ℂ => ((p.1, z), p.2)) by fun_prop) have he (x : ℂ) : pairMass ν ρ δ x z = ∫⁻ y : ℂ, A.indicator 1 (x, y) ∂ν := by rw [show (fun y : ℂ => A.indicator 1 (x, y)) = {y : ℂ | pairCondition ρ δ x z y}.indicator 1 by ext y; exact (indicator_comp_right (fun y : ℂ => (x, y))).symm] exact (lintegral_indicator_one ((measurableSet_pairCondition ρ δ).preimage (show Measurable (fun y : ℂ => ((x, z), y)) by fun_prop))).symm simp_rw [he] rw [lintegral_lintegral_swap (f := fun x y : ℂ => A.indicator 1 (x, y)) (measurable_const.indicator hA).aemeasurable] calc _ ≤ ∫⁻ y : ℂ, ENNReal.ofReal (156 * δ / ρ) ∂ν := by apply lintegral_mono intro y have hei : (fun x : ℂ => A.indicator (1 : ℂ × ℂ → ℝ≥0∞) (x, y)) = {x : ℂ | pairCondition ρ δ x z y}.indicator 1 := by ext x exact (indicator_comp_right (fun x : ℂ => (x, y))).symm dsimp only rw [hei] have hsm : MeasurableSet {x : ℂ | pairCondition ρ δ x z y} := (measurableSet_pairCondition ρ δ).preimage (show Measurable (fun x : ℂ => ((x, z), y)) by fun_prop) exact (lintegral_indicator_one hsm).le.trans (volume_pairCondition hρ hδ hδquarter z y) _ = _ := lintegral_const _ /-- The second contribution is controlled by monotone ray jumps and polar coordinates. -/ theorem lintegral_rayIndicator (ν S : Measure ℂ) [IsFiniteMeasure ν] [IsFiniteMeasure S] (ρ : ℝ) {T U v : ℝ} (hT : 0 ≤ T) (hU : 0 < U) (hv : 0 ≤ v) : (∫⁻ x : ℂ, ∫⁻ z : ℂ, rayIndicator S ρ T U v x z ∂ν) ≤ (ENNReal.ofReal (6 * Real.pi) * ENNReal.ofReal (2 * v * (1 + T / U))) * ν univ := by rw [lintegral_lintegral_swap (measurable_rayIndicator S ρ T U v).aemeasurable] calc _ ≤ ∫⁻ _z : ℂ, ENNReal.ofReal (6 * Real.pi) * ENNReal.ofReal (2 * v * (1 + T / U)) ∂ν := by apply lintegral_mono intro z have he : (fun x : ℂ => rayIndicator S ρ T U v x z) = {x : ℂ | rayCondition S ρ T U v x z}.indicator 1 := by ext x exact (indicator_comp_right (fun x : ℂ => (x, z))).symm dsimp only rw [he] have hsm : MeasurableSet {x : ℂ | rayCondition S ρ T U v x z} := (measurableSet_rayCondition S ρ T U v).preimage (show Measurable (fun x : ℂ => (x, z)) by fun_prop) exact (lintegral_indicator_one hsm).le.trans (volume_rayCondition S ρ hT hU hv z) _ = _ := lintegral_const _ end DiskGeometry namespace DiskGeometry /-- Failure of every cap forces the majorant bracket to be at least one. -/ theorem bad_cap_bracket (S ν : Measure ℂ) [IsFiniteMeasure S] [IsFiniteMeasure ν] {x z : ℂ} {r ρ δ T U : ℝ} (hr : r ∈ Icc (1 : ℝ) 2) (hρ : 0 < ρ) (hρ1 : ρ ≤ 1) (hδ : 0 < δ) (hδρ : δ ≤ ρ ^ 2 / 1600) (hT : 0 ≤ T) (hU : 0 < U) (hz : z ∈ thinShell x r δ) (hi : S (Metric.ball x (r - δ)) ≤ ENNReal.ofReal T) (hb : ENNReal.ofReal (2 * U) < (S + ν) (thinShell x r δ \ Metric.ball z ρ)) : 1 ≤ (ENNReal.ofReal U)⁻¹ * pairMass ν ρ δ x z + rayIndicator S ρ T U (200 * δ / ρ ^ 2) x z := by have hδquarter : δ ≤ 1 / 4 := by nlinarith obtain ⟨hl, hl3⟩ := shell_center_bounds hr hδquarter hz have hcases : ENNReal.ofReal U ≤ ν (thinShell x r δ \ Metric.ball z ρ) ∨ ENNReal.ofReal U ≤ S (thinShell x r δ \ Metric.ball z ρ) := by by_contra h push Not at h have he2 : ENNReal.ofReal (2 * U) = ENNReal.ofReal U + ENNReal.ofReal U := by rw [two_mul, ENNReal.ofReal_add hU.le hU.le] have hm : (S + ν) (thinShell x r δ \ Metric.ball z ρ) ≤ ENNReal.ofReal (2 * U) := by rw [Measure.add_apply, he2] exact add_le_add h.2.le h.1.le exact (not_le_of_gt hb) hm rcases hcases with hν | hS · have hp : ENNReal.ofReal U ≤ pairMass ν ρ δ x z := by apply hν.trans (measure_mono ?_) rintro y ⟨hy, hsep⟩ exact ⟨hl, hl3, shell_pair_close hy hz, le_of_not_gt (by simpa only [Metric.mem_ball, dist_eq_norm] using hsep)⟩ calc 1 = (ENNReal.ofReal U)⁻¹ * ENNReal.ofReal U := (ENNReal.inv_mul_cancel (ENNReal.ofReal_pos.mpr hU).ne' ENNReal.ofReal_ne_top).symm _ ≤ (ENNReal.ofReal U)⁻¹ * pairMass ν ρ δ x z := mul_le_mul_right hp _ _ ≤ _ := le_self_add · have hi' : (S (Metric.ball x (r - δ))).toReal ≤ T := by have := ENNReal.toReal_mono ENNReal.ofReal_ne_top hi simpa only [ENNReal.toReal_ofReal hT] using this have hS' : U ≤ (S (thinShell x r δ \ Metric.ball z ρ)).toReal := (ENNReal.ofReal_le_iff_le_toReal (measure_ne_top S _)).1 hS have hc : rayCondition S ρ T U (200 * δ / ρ ^ 2) x z := ⟨hl, hl3, (earlier_mass_le_interior S hr hρ hρ1 hδ hδρ hz).trans hi', hS'.trans (shell_mass_le_jump S hr hρ hρ1 hδ hδρ hz)⟩ have hone : rayIndicator S ρ T U (200 * δ / ρ ^ 2) x z = 1 := indicator_of_mem hc 1 rw [hone] exact le_add_self /-- A radius-independent Borel majorant combining separated shell pairs and ray jumps. -/ def shellMajorant (ν S : Measure ℂ) (ρ δ T U : ℝ) (x : ℂ) : ℝ≥0∞ := (ENNReal.ofReal U)⁻¹ ^ 2 * (∫⁻ z, pairMass ν ρ δ x z ∂ν) + (ENNReal.ofReal U)⁻¹ * (∫⁻ z, rayIndicator S ρ T U (200 * δ / ρ ^ 2) x z ∂ν) theorem measurable_shellMajorant (ν S : Measure ℂ) [IsFiniteMeasure ν] [IsFiniteMeasure S] (ρ δ T U : ℝ) : Measurable (shellMajorant ν S ρ δ T U) := by exact ((measurable_pairMass ν ρ δ).lintegral_prod_right.const_mul _).add ((measurable_rayIndicator S ρ T U (200 * δ / ρ ^ 2)).lintegral_prod_right.const_mul _) theorem shellMajorant_eq_bracket (ν S : Measure ℂ) [IsFiniteMeasure ν] [IsFiniteMeasure S] (ρ δ T U : ℝ) (x : ℂ) : shellMajorant ν S ρ δ T U x = (ENNReal.ofReal U)⁻¹ * ∫⁻ z, ((ENNReal.ofReal U)⁻¹ * pairMass ν ρ δ x z + rayIndicator S ρ T U (200 * δ / ρ ^ 2) x z) ∂ν := by have hp : Measurable (fun z : ℂ => pairMass ν ρ δ x z) := (measurable_pairMass ν ρ δ).comp (measurable_const.prodMk measurable_id) rw [lintegral_add_left (hp.const_mul _), lintegral_const_mul _ hp] unfold shellMajorant ring /-- The majorant is integrable uniformly over all radii. -/ theorem lintegral_shellMajorant (ν S : Measure ℂ) [IsFiniteMeasure ν] [IsFiniteMeasure S] {ρ δ T U : ℝ} (hρ : 0 < ρ) (hδ : 0 ≤ δ) (hδquarter : δ ≤ 1 / 4) (hT : 0 ≤ T) (hU : 0 < U) : (∫⁻ x, shellMajorant ν S ρ δ T U x) ≤ (ENNReal.ofReal U)⁻¹ ^ 2 * (ENNReal.ofReal (156 * δ / ρ) * ν univ * ν univ) + (ENNReal.ofReal U)⁻¹ * ((ENNReal.ofReal (6 * Real.pi) * ENNReal.ofReal (2 * (200 * δ / ρ ^ 2) * (1 + T / U))) * ν univ) := by have hp : Measurable (fun x : ℂ => ∫⁻ z, pairMass ν ρ δ x z ∂ν) := (measurable_pairMass ν ρ δ).lintegral_prod_right have hr : Measurable (fun x : ℂ => ∫⁻ z, rayIndicator S ρ T U (200 * δ / ρ ^ 2) x z ∂ν) := (measurable_rayIndicator S ρ T U (200 * δ / ρ ^ 2)).lintegral_prod_right simp only [shellMajorant] rw [lintegral_add_left (hp.const_mul _), lintegral_const_mul _ hp, lintegral_const_mul _ hr] exact add_le_add (mul_le_mul_right (lintegral_pairMass ν hρ hδ hδquarter) _) (mul_le_mul_right (lintegral_rayIndicator ν S ρ hT hU (by positivity)) _) /-- The majorant detects simultaneous failure of concentration at every cap. -/ theorem one_le_majorant_of_no_cap (ν S : Measure ℂ) [IsFiniteMeasure ν] [IsFiniteMeasure S] {ρ δ T U : ℝ} (hρ : 0 < ρ) (hρ1 : ρ ≤ 1) (hδ : 0 < δ) (hδρ : δ ≤ ρ ^ 2 / 1600) (hT : 0 ≤ T) (hU : 0 < U) {x : ℂ} {r : ℝ} (hr : r ∈ Icc (1 : ℝ) 2) (hν : ENNReal.ofReal U ≤ ν (thinShell x r δ)) (hS : S (Metric.ball x (r - δ)) ≤ ENNReal.ofReal T) (hbad : ∀ z ∈ thinShell x r δ, ENNReal.ofReal (2 * U) < (S + ν) (thinShell x r δ \ Metric.ball z ρ)) : 1 ≤ shellMajorant ν S ρ δ T U x := by have hu0 : ENNReal.ofReal U ≠ 0 := (ENNReal.ofReal_pos.mpr hU).ne' rw [shellMajorant_eq_bracket] calc 1 = (ENNReal.ofReal U)⁻¹ * ENNReal.ofReal U := (ENNReal.inv_mul_cancel hu0 ENNReal.ofReal_ne_top).symm _ ≤ (ENNReal.ofReal U)⁻¹ * ν (thinShell x r δ) := mul_le_mul_right hν _ _ = (ENNReal.ofReal U)⁻¹ * (∫⁻ z, (thinShell x r δ).indicator 1 z ∂ν) := by rw [lintegral_indicator_one (measurableSet_thinShell x r δ)] _ ≤ _ := by apply mul_le_mul_right apply lintegral_mono intro z by_cases hz : z ∈ thinShell x r δ · rw [indicator_of_mem hz] exact bad_cap_bracket S ν hr hρ hρ1 hδ hδρ hT hU hz hS (hbad z hz) · rw [indicator_of_notMem hz] exact bot_le /-- Full radius-independent shell concentration, with explicit absolute constants. -/ theorem shell_concentration (ν S : Measure ℂ) [IsFiniteMeasure ν] [IsFiniteMeasure S] {ρ δ T U : ℝ} (hρ : 0 < ρ) (hρ1 : ρ ≤ 1) (hδ : 0 < δ) (hδρ : δ ≤ ρ ^ 2 / 1600) (hT : 0 ≤ T) (hU : 0 < U) : ∃ E : Set ℂ, MeasurableSet E ∧ volume E ≤ (ENNReal.ofReal U)⁻¹ ^ 2 * (ENNReal.ofReal (156 * δ / ρ) * ν univ * ν univ) + (ENNReal.ofReal U)⁻¹ * ((ENNReal.ofReal (6 * Real.pi) * ENNReal.ofReal (2 * (200 * δ / ρ ^ 2) * (1 + T / U))) * ν univ) ∧ ∀ x ∉ E, ∀ r ∈ Icc (1 : ℝ) 2, ENNReal.ofReal U ≤ ν (thinShell x r δ) → S (Metric.ball x (r - δ)) ≤ ENNReal.ofReal T → ∃ z ∈ thinShell x r δ, (S + ν) (thinShell x r δ \ Metric.ball z ρ) ≤ ENNReal.ofReal (2 * U) := by let E : Set ℂ := {x | 1 ≤ shellMajorant ν S ρ δ T U x} have hE : MeasurableSet E := measurableSet_le measurable_const (measurable_shellMajorant ν S ρ δ T U) refine ⟨E, hE, ?_, ?_⟩ · have hm := meas_ge_le_lintegral_div (μ := (volume : Measure ℂ)) (measurable_shellMajorant ν S ρ δ T U).aemeasurable one_ne_zero ENNReal.one_ne_top simp only [div_one] at hm refine hm.trans (lintegral_shellMajorant ν S hρ hδ.le ?_ hT hU) have hρsq : ρ ^ 2 ≤ 1 := by nlinarith linarith · intro x hx r hr hν hS by_contra h push Not at h exact hx (one_le_majorant_of_no_cap ν S hρ hρ1 hδ hδρ hT hU hr hν hS h) end DiskGeometry open MeasureTheory Set Filter open scoped Topology RealInnerProductSpace ContDiff noncomputable section end end end OAI