import OAI.Probability.GaussianReplacement.Basic namespace OAI open MeasureTheory ProbabilityTheory Set Filter open scoped ENNReal NNReal RealInnerProductSpace Topology noncomputable section namespace CurrentProjection section InverseMoments variable {X : Type*} [MeasurableSpace X] omit [MeasurableSpace X] in lemma inverse_pow_dyadic_majorant (f : X → ℝ) (m : ℕ) (hf : ∀ x, 0 ≤ f x) (x : X) : ENNReal.ofReal ((f x)⁻¹ ^ m) ≤ 1 + ∑' k : ℕ, ({x | f x ≤ (1 / 2 : ℝ) ^ k}).indicator (fun _ => ENNReal.ofReal ((2 : ℝ) ^ (m * (k + 1)))) x := by classical by_cases hzero : f x = 0 · simp only [hzero, inv_zero] exact le_add_of_le_left (by cases m <;> simp) have hxpos : 0 < f x := lt_of_le_of_ne (hf x) (Ne.symm hzero) by_cases hxone : 1 ≤ f x · apply le_add_of_le_left have h : (f x)⁻¹ ^ m ≤ 1 := pow_le_one₀ (inv_nonneg.mpr (hf x)) (inv_le_one_of_one_le₀ hxone) simpa using ENNReal.ofReal_le_ofReal h obtain ⟨k, hklo, hkhi⟩ := exists_nat_pow_near_of_lt_one hxpos (le_of_not_ge hxone) (by norm_num : (0 : ℝ) < 1 / 2) (by norm_num : (1 / 2 : ℝ) < 1) have hinv : (f x)⁻¹ ≤ (2 : ℝ) ^ (k + 1) := by have h := (inv_le_inv₀ hxpos (pow_pos (by norm_num : (0 : ℝ) < 1 / 2) _)).2 hklo.le simpa using h have hpow : ENNReal.ofReal ((f x)⁻¹ ^ m) ≤ ENNReal.ofReal ((2 : ℝ) ^ (m * (k + 1))) := by apply ENNReal.ofReal_le_ofReal simpa [← pow_mul, Nat.mul_comm] using pow_le_pow_left₀ (inv_nonneg.mpr (hf x)) hinv m calc ENNReal.ofReal ((f x)⁻¹ ^ m) ≤ ENNReal.ofReal ((2 : ℝ) ^ (m * (k + 1))) := hpow _ = ({x | f x ≤ (1 / 2 : ℝ) ^ k}).indicator (fun _ => ENNReal.ofReal ((2 : ℝ) ^ (m * (k + 1)))) x := by rw [Set.indicator_of_mem (show x ∈ {x | f x ≤ (1 / 2 : ℝ) ^ k} from hkhi)] _ ≤ ∑' j : ℕ, ({x | f x ≤ (1 / 2 : ℝ) ^ j}).indicator (fun _ => ENNReal.ofReal ((2 : ℝ) ^ (m * (j + 1)))) x := ENNReal.le_tsum (f := fun j => ({x | f x ≤ (1 / 2 : ℝ) ^ j}).indicator (fun _ => ENNReal.ofReal ((2 : ℝ) ^ (m * (j + 1)))) x) k _ ≤ 1 + ∑' j : ℕ, ({x | f x ≤ (1 / 2 : ℝ) ^ j}).indicator (fun _ => ENNReal.ofReal ((2 : ℝ) ^ (m * (j + 1)))) x := le_add_left le_rfl lemma dyadic_power_cancel (m k : ℕ) : (2 : ℝ) ^ (m * (k + 1)) * ((1 / 2 : ℝ) ^ k) ^ (m + 1) = (2 : ℝ) ^ m * (1 / 2 : ℝ) ^ k := by rw [Nat.mul_add, mul_one, pow_add, pow_succ] have h : (2 : ℝ) ^ (m * k) * ((1 / 2 : ℝ) ^ k) ^ m = 1 := by rw [Nat.mul_comm m k, pow_mul, ← mul_pow, ← mul_pow] norm_num calc 2 ^ (m * k) * 2 ^ m * (((1 / 2 : ℝ) ^ k) ^ m * (1 / 2 : ℝ) ^ k) = (2 ^ (m * k) * ((1 / 2 : ℝ) ^ k) ^ m) * (2 ^ m * (1 / 2 : ℝ) ^ k) := by ring _ = _ := by rw [h, one_mul] theorem inverse_moment_of_tube_bound (μ : Measure X) (f : X → ℝ) (m : ℕ) (hf : Measurable f) (hnonneg : ∀ x, 0 ≤ f x) (C : ℝ≥0∞) (htube : ∀ r : ℝ, 0 < r → r ≤ 1 → μ {x | f x ≤ r} ≤ C * ENNReal.ofReal (r ^ (m + 1))) : ∫⁻ x, ENNReal.ofReal ((f x)⁻¹ ^ m) ∂μ ≤ μ Set.univ + C * (2 : ℝ≥0∞) ^ (m + 1) := by classical let g : ℕ → X → ℝ≥0∞ := fun k => ({x | f x ≤ (1 / 2 : ℝ) ^ k}).indicator (fun _ => ENNReal.ofReal ((2 : ℝ) ^ (m * (k + 1)))) have hg : ∀ k, Measurable (g k) := fun k => measurable_const.indicator (measurableSet_le hf measurable_const) calc ∫⁻ x, ENNReal.ofReal ((f x)⁻¹ ^ m) ∂μ ≤ ∫⁻ x, 1 + ∑' k, g k x ∂μ := lintegral_mono (inverse_pow_dyadic_majorant f m hnonneg) _ = μ Set.univ + ∑' k, ∫⁻ x, g k x ∂μ := by rw [lintegral_add_left measurable_const, lintegral_const, one_mul, lintegral_tsum (fun k => (hg k).aemeasurable)] _ ≤ μ Set.univ + ∑' k, C * ENNReal.ofReal ((2 : ℝ) ^ m * (1 / 2 : ℝ) ^ k) := by apply add_le_add le_rfl apply ENNReal.tsum_le_tsum intro k dsimp only [g] rw [lintegral_indicator (measurableSet_le hf measurable_const), lintegral_const, Measure.restrict_apply_univ] calc ENNReal.ofReal ((2 : ℝ) ^ (m * (k + 1))) * μ {x | f x ≤ (1 / 2 : ℝ) ^ k} ≤ ENNReal.ofReal ((2 : ℝ) ^ (m * (k + 1))) * (C * ENNReal.ofReal (((1 / 2 : ℝ) ^ k) ^ (m + 1))) := by gcongr exact htube _ (by positivity) (pow_le_one₀ (by norm_num) (by norm_num)) _ = _ := by rw [← mul_assoc, mul_comm _ C, mul_assoc, ← ENNReal.ofReal_mul (by positivity), dyadic_power_cancel] _ = μ Set.univ + C * (2 : ℝ≥0∞) ^ (m + 1) := by simp_rw [ENNReal.ofReal_mul (by positivity : (0 : ℝ) ≤ 2 ^ m), ENNReal.ofReal_pow (by norm_num : (0 : ℝ) ≤ 1 / 2), ENNReal.ofReal_pow (by norm_num : (0 : ℝ) ≤ 2)] rw [ENNReal.tsum_mul_left, ENNReal.tsum_mul_left, ENNReal.tsum_geometric] norm_num [pow_succ, mul_assoc, ENNReal.ofReal_div_of_pos] end InverseMoments section FrostmanGeometry variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] def HasLocalMassBound (ν : Measure E) (a : ℝ) : Prop := ∀ t ∈ ν.support, ∀ r : ℝ, 0 < r → ν (Metric.closedBall t r) ≤ ENNReal.ofReal (r ^ a) omit [BorelSpace E] in theorem HasLocalMassBound.any_center {ν : Measure E} {a : ℝ} (hν : HasLocalMassBound ν a) (z : E) {r : ℝ} (hr : 0 < r) : ν (Metric.closedBall z r) ≤ ENNReal.ofReal ((2 * r) ^ a) := by by_cases hmass : ν (Metric.closedBall z r) = 0 · simp [hmass] obtain ⟨t, htball, htsupp⟩ := ν.nonempty_inter_support_of_pos (pos_iff_ne_zero.mpr hmass) refine (measure_mono ?_).trans (hν t htsupp (2 * r) (by positivity)) intro x hx apply Metric.mem_closedBall.mpr calc dist x t ≤ dist x z + dist z t := dist_triangle _ _ _ _ ≤ r + r := add_le_add (Metric.mem_closedBall.mp hx) (by simpa [dist_comm] using Metric.mem_closedBall.mp htball) _ = 2 * r := by ring theorem card_le_volume_ratio (s : Finset E) {r R : ℝ} (hr : 0 < r) (hR : 0 ≤ R) (hs : ∀ x ∈ s, ‖x‖ ≤ R) (hsep : ∀ x ∈ s, ∀ y ∈ s, x ≠ y → r ≤ dist x y) : (s.card : ℝ) ≤ ((R + r / 2) / (r / 2)) ^ Module.finrank ℝ E := by classical let ν : Measure E := Measure.addHaar let δ : ℝ := r / 2 let ρ : ℝ := R + r / 2 have hδ : 0 < δ := by dsimp [δ]; positivity have hρ : 0 < ρ := by dsimp [ρ]; positivity let A : Set E := ⋃ c ∈ s, Metric.ball c δ have hd : Set.Pairwise (s : Set E) (fun c d => Disjoint (Metric.ball c δ) (Metric.ball d δ)) := by intro c hc d hd hcd apply Metric.ball_disjoint_ball simpa [δ] using hsep c hc d hd hcd have hsub : A ⊆ Metric.ball (0 : E) ρ := by refine iUnion₂_subset fun x hx => ?_ apply Metric.ball_subset_ball' dsimp [δ, ρ] rw [dist_zero_right] linarith [hs x hx] have I : (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ Module.finrank ℝ E) * ν (Metric.ball 0 1) ≤ ENNReal.ofReal (ρ ^ Module.finrank ℝ E) * ν (Metric.ball 0 1) := by calc (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ Module.finrank ℝ E) * ν (Metric.ball 0 1) = ν A := by dsimp only [A] rw [measure_biUnion_finset hd (fun _ _ => Metric.isOpen_ball.measurableSet)] simp only [ν.addHaar_ball_of_pos _ hδ, Finset.sum_const, nsmul_eq_mul, mul_assoc] _ ≤ ν (Metric.ball (0 : E) ρ) := measure_mono hsub _ = _ := ν.addHaar_ball_of_pos 0 hρ have J : (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ Module.finrank ℝ E) ≤ ENNReal.ofReal (ρ ^ Module.finrank ℝ E) := (ENNReal.mul_le_mul_iff_left (Metric.measure_ball_pos ν 0 zero_lt_one).ne' measure_ball_lt_top.ne).mp I have K : (s.card : ℝ) * δ ^ Module.finrank ℝ E ≤ ρ ^ Module.finrank ℝ E := by simpa only [ENNReal.toReal_mul, ENNReal.toReal_natCast, ENNReal.toReal_ofReal (pow_nonneg hδ.le _)] using ENNReal.toReal_le_of_le_ofReal (pow_nonneg hρ.le _) J rw [div_pow] exact (le_div_iff₀ (pow_pos hδ _)).mpr K end FrostmanGeometry section FiniteNets variable {E : Type*} [PseudoMetricSpace E] theorem exists_net_of_finite_packing_bound (A : Set E) {r : ℝ} (hr : 0 ≤ r) (N : ℕ) (hcard : ∀ s : Finset E, (∀ x ∈ s, x ∈ A) → (∀ x ∈ s, ∀ y ∈ s, x ≠ y → r < dist x y) → s.card ≤ N) : ∃ s : Finset E, (∀ x ∈ s, x ∈ A) ∧ (∀ x ∈ s, ∀ y ∈ s, x ≠ y → r < dist x y) ∧ ∀ x ∈ A, ∃ y ∈ s, dist x y ≤ r := by classical let S : Set ℕ := {n | ∃ s : Finset E, (∀ x ∈ s, x ∈ A) ∧ (∀ x ∈ s, ∀ y ∈ s, x ≠ y → r < dist x y) ∧ s.card = n} have hS : S.Finite := (Set.finite_Iic N).subset (by rintro n ⟨s, hs, hsep, rfl⟩ exact hcard s hs hsep) have hS0 : S.Nonempty := ⟨0, ∅, by simp⟩ obtain ⟨n, hn, hmax⟩ := Set.exists_max_image S id hS hS0 obtain ⟨s, hs, hsep, rfl⟩ := hn refine ⟨s, hs, hsep, ?_⟩ intro x hx by_contra! hnot have hxs : x ∉ s := by intro h have := hnot x h simpa using lt_of_le_of_lt hr this have hins : ∀ y ∈ insert x s, y ∈ A := by intro y hy rcases Finset.mem_insert.mp hy with rfl | hy · exact hx · exact hs y hy have hsep' : ∀ y ∈ insert x s, ∀ z ∈ insert x s, y ≠ z → r < dist y z := by intro y hy z hz hne rcases Finset.mem_insert.mp hy with hxy | hys · subst y rcases Finset.mem_insert.mp hz with hxz | hzs · exact (hne hxz.symm).elim · exact hnot z hzs · rcases Finset.mem_insert.mp hz with hxz | hzs · subst z simpa [dist_comm] using hnot y hys · exact hsep y hys z hzs hne have hinS : (insert x s).card ∈ S := ⟨insert x s, hins, hsep', rfl⟩ have hh := hmax _ hinS simp only [id_eq, Finset.card_insert_of_notMem hxs] at hh omega end FiniteNets section EuclideanNets variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] theorem exists_unit_ball_net {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) : ∃ s : Finset E, (∀ x ∈ s, ‖x‖ ≤ 1) ∧ (s.card : ℝ) ≤ (3 / r) ^ Module.finrank ℝ E ∧ ∀ x : E, ‖x‖ ≤ 1 → ∃ y ∈ s, dist x y ≤ r := by classical have hbound : ∀ s : Finset E, (∀ x ∈ s, ‖x‖ ≤ 1) → (∀ x ∈ s, ∀ y ∈ s, x ≠ y → r < dist x y) → (s.card : ℝ) ≤ (3 / r) ^ Module.finrank ℝ E := by intro s hs hsep refine (card_le_volume_ratio s hr zero_le_one hs (fun x hx y hy hne => (hsep x hx y hy hne).le)).trans ?_ apply pow_le_pow_left₀ (by positivity) apply (div_le_div_iff₀ (by positivity : 0 < r / 2) hr).mpr nlinarith obtain ⟨s, hs, hsep, hnet⟩ := exists_net_of_finite_packing_bound {x : E | ‖x‖ ≤ 1} hr.le ⌈(3 / r) ^ Module.finrank ℝ E⌉₊ (by intro s hs hsep exact_mod_cast (hbound s hs hsep).trans (Nat.le_ceil _)) exact ⟨s, hs, hbound s hs hsep, hnet⟩ end EuclideanNets section AffineTubes variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] open EuclideanGeometry theorem exists_affine_tube_cover (L : AffineSubspace ℝ E) [Nonempty L] {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) : ∃ s : Finset L.direction, (s.card : ℝ) ≤ (3 / r) ^ Module.finrank ℝ L.direction ∧ ∀ t : E, ‖t‖ ≤ 1 → Metric.infDist t L ≤ r → ∃ y ∈ s, dist t ((y : E) + (orthogonalProjection L 0 : E)) ≤ 2 * r := by classical obtain ⟨s, _, hcard, hnet⟩ := exists_unit_ball_net (E := L.direction) hr hr1 refine ⟨s, hcard, ?_⟩ intro t ht hdist obtain ⟨y, hy, hty⟩ := hnet (L.direction.orthogonalProjectionOnto t) ((L.direction.norm_orthogonalProjectionOnto_apply_le t).trans ht) refine ⟨y, hy, ?_⟩ have hproj : (orthogonalProjection L t : E) = (L.direction.orthogonalProjectionOnto t : E) + (orthogonalProjection L 0 : E) := by have h := (orthogonalProjection L).map_vadd (0 : E) t simpa only [vadd_eq_add, add_zero, orthogonalProjection_contLinear, AffineSubspace.coe_vadd] using congrArg (fun x : L => (x : E)) h have hpdist : dist (orthogonalProjection L t : E) ((y : E) + (orthogonalProjection L 0 : E)) ≤ r := by rw [hproj, dist_add_right] exact hty have hd : dist t (orthogonalProjection L t : E) ≤ r := by rwa [dist_orthogonalProjection_eq_infDist] calc dist t ((y : E) + (orthogonalProjection L 0 : E)) ≤ dist t (orthogonalProjection L t : E) + dist (orthogonalProjection L t : E) ((y : E) + (orthogonalProjection L 0 : E)) := dist_triangle _ _ _ _ ≤ r + r := add_le_add hd hpdist _ = 2 * r := by ring theorem HasLocalMassBound.affine_tube {ν : Measure E} {a : ℝ} (hν : HasLocalMassBound ν a) (hsupport : ν.support ⊆ Metric.closedBall 0 1) (L : AffineSubspace ℝ E) [Nonempty L] {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) : ν {t | Metric.infDist t L ≤ r} ≤ ENNReal.ofReal ((3 / r) ^ Module.finrank ℝ L.direction * (4 * r) ^ a) := by classical obtain ⟨s, hcard, hcover⟩ := exists_affine_tube_cover L hr hr1 let c : L.direction → E := fun y => (y : E) + (orthogonalProjection L 0 : E) have hsub : {t | Metric.infDist t L ≤ r} ≤ᵐ[ν] ⋃ y ∈ s, Metric.closedBall (c y) (2 * r) := by filter_upwards [ν.support_mem_ae] with t ht intro hdist have htunit : ‖t‖ ≤ 1 := by simpa using hsupport ht obtain ⟨y, hy, hty⟩ := hcover t htunit hdist exact mem_iUnion₂.mpr ⟨y, hy, hty⟩ calc ν {t | Metric.infDist t L ≤ r} ≤ ν (⋃ y ∈ s, Metric.closedBall (c y) (2 * r)) := measure_mono_ae hsub _ ≤ ∑ y ∈ s, ν (Metric.closedBall (c y) (2 * r)) := measure_biUnion_finset_le _ _ _ ≤ ∑ _y ∈ s, ENNReal.ofReal ((4 * r) ^ a) := by refine Finset.sum_le_sum fun y _ => ?_ simpa only [show (2 : ℝ) * (2 * r) = 4 * r by ring] using hν.any_center (c y) (r := 2 * r) (by positivity) _ = (s.card : ℝ≥0∞) * ENNReal.ofReal ((4 * r) ^ a) := by simp _ ≤ ENNReal.ofReal ((3 / r) ^ Module.finrank ℝ L.direction) * ENNReal.ofReal ((4 * r) ^ a) := by gcongr exact_mod_cast ENNReal.ofReal_le_ofReal hcard _ = _ := (ENNReal.ofReal_mul (by positivity)).symm lemma tube_radius_factors (ℓ : ℕ) (a : ℝ) {r : ℝ} (hr : 0 < r) : (3 / r) ^ ℓ * (4 * r) ^ a = (3 : ℝ) ^ ℓ * (4 : ℝ) ^ a * r ^ (a - ℓ) := by rw [div_pow, Real.mul_rpow (by norm_num) hr.le, Real.rpow_sub_natCast hr.ne'] ring theorem HasLocalMassBound.affine_tube_pow {ν : Measure E} {a : ℝ} (hν : HasLocalMassBound ν a) (hsupport : ν.support ⊆ Metric.closedBall 0 1) (L : AffineSubspace ℝ E) [Nonempty L] (k : ℕ) (hmargin : (Module.finrank ℝ L.direction : ℝ) + k ≤ a) {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) : ν {t | Metric.infDist t L ≤ r} ≤ ENNReal.ofReal ((3 : ℝ) ^ Module.finrank ℝ L.direction * (4 : ℝ) ^ a) * ENNReal.ofReal (r ^ k) := by refine (hν.affine_tube hsupport L hr hr1).trans ?_ rw [tube_radius_factors _ _ hr, ← ENNReal.ofReal_mul (by positivity)] apply ENNReal.ofReal_le_ofReal apply mul_le_mul_of_nonneg_left _ (by positivity) rw [← Real.rpow_natCast] exact Real.rpow_le_rpow_of_exponent_ge hr hr1 (by linarith) theorem HasLocalMassBound.affine_inverse_moment {ν : Measure E} {a : ℝ} (hν : HasLocalMassBound ν a) (hsupport : ν.support ⊆ Metric.closedBall 0 1) (L : AffineSubspace ℝ E) [Nonempty L] (m : ℕ) (hmargin : (Module.finrank ℝ L.direction : ℝ) + (m + 1 : ℕ) ≤ a) : ∫⁻ t, ENNReal.ofReal ((Metric.infDist t L)⁻¹ ^ m) ∂ν ≤ ν Set.univ + ENNReal.ofReal ((3 : ℝ) ^ Module.finrank ℝ L.direction * (4 : ℝ) ^ a) * (2 : ℝ≥0∞) ^ (m + 1) := by apply inverse_moment_of_tube_bound ν _ m (Metric.continuous_infDist_pt _).measurable (fun _ => Metric.infDist_nonneg) intro r hr hr1 exact hν.affine_tube_pow hsupport L (m + 1) hmargin hr hr1 theorem HasLocalMassBound.affine_null {ν : Measure E} {a : ℝ} (hν : HasLocalMassBound ν a) (hsupport : ν.support ⊆ Metric.closedBall 0 1) (L : AffineSubspace ℝ E) [Nonempty L] (hmargin : (Module.finrank ℝ L.direction : ℝ) + 1 ≤ a) : ν L = 0 := by let C : ℝ := (3 : ℝ) ^ Module.finrank ℝ L.direction * (4 : ℝ) ^ a have hC : 0 ≤ C := by dsimp [C]; positivity have hb (n : ℕ) : ν L ≤ ENNReal.ofReal (C * (1 / 2 : ℝ) ^ n) := by have hn : 0 < (1 / 2 : ℝ) ^ n := by positivity have hn1 : (1 / 2 : ℝ) ^ n ≤ 1 := pow_le_one₀ (by norm_num) (by norm_num) calc ν L ≤ ν {t | Metric.infDist t L ≤ (1 / 2 : ℝ) ^ n} := by apply measure_mono intro t ht simp only [mem_ofPred_eq, Metric.infDist_zero_of_mem ht] exact hn.le _ ≤ ENNReal.ofReal C * ENNReal.ofReal (((1 / 2 : ℝ) ^ n) ^ 1) := hν.affine_tube_pow hsupport L 1 (by simpa using hmargin) hn hn1 _ = _ := by rw [pow_one, ENNReal.ofReal_mul hC] have hlim : Tendsto (fun n : ℕ => ENNReal.ofReal (C * (1 / 2 : ℝ) ^ n)) atTop (𝓝 0) := by convert ENNReal.tendsto_ofReal ((tendsto_const_nhds (x := C)).mul (tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0 : ℝ) ≤ 1 / 2) (by norm_num : (1 / 2 : ℝ) < 1))) using 1 simp exact le_antisymm (ge_of_tendsto' hlim hb) (by positivity) end AffineTubes end CurrentProjection end end OAI