| import OAI.Geometry.RuledCovering.ConeVolume |
| import OAI.Geometry.RuledCovering.SingularOverlap |
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| namespace OAI |
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| noncomputable section |
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| open Set MeasureTheory Filter Metric |
| open scoped Topology ContDiff Pointwise |
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| namespace RuledApproximation |
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| lemma positivepart_increment {a h t : ℝ} (hh : 0 ≤ h) : |
| 0 ≤ max (a+h-t) 0-max (a-t) 0 ∧ max (a+h-t) 0-max (a-t) 0 ≤ h := by |
| by_cases ht : t ≤ a |
| · rw [max_eq_left (by linarith),max_eq_left (by linarith)] |
| constructor <;> linarith |
| · rw [show max (a-t) 0 = 0 from max_eq_right (by linarith), sub_zero] |
| exact ⟨le_max_right _ _, max_le (by linarith) hh⟩ |
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| lemma positivepart_floor_increment {a d h t : ℝ} (hd : d ≤ a) (hh : 0 ≤ h) : |
| max (a+h-max d t) 0-max (a-max d t) 0 = max (a+h-t) 0-max (a-t) 0 := by |
| by_cases ht : t ≤ d |
| · rw [max_eq_left ht,max_eq_left (by linarith),max_eq_left (by linarith), |
| max_eq_left (by linarith),max_eq_left (by linarith)] |
| ring |
| · rw [max_eq_right (le_of_not_ge ht)] |
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| lemma rpow_five_halves_div_square {c : ℝ} (hc : 0 < c) : |
| c^(5/2:ℝ)/c^2=c^(1/2:ℝ) := by |
| rw [← Real.rpow_natCast,← Real.rpow_sub hc] |
| norm_num |
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| def capIncrementQuotient (p c b : ℝ) : ℝ := |
| (max ((1/2:ℝ)+c^2*ceilingPolynomial p b+c^(5/2:ℝ)-|b|) 0 - |
| max ((1/2:ℝ)-|b|) 0)/c^2 |
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| lemma capIncrementQuotient_nonneg_bound {p c b : ℝ} (hp : 0 ≤ p) (hp2 : p ≤ 2) (hc : 0 < c) : |
| 0 ≤ capIncrementQuotient p c b ∧ |
| capIncrementQuotient p c b ≤ ceilingPolynomial p b+c^(1/2:ℝ) := by |
| have hd := ceiling_nonneg hp hp2 b |
| have hi := positivepart_increment (a := (1/2:ℝ)) (t := |b|) |
| (h := c^2*ceilingPolynomial p b+c^(5/2:ℝ)) (by positivity) |
| have he : (c^2*ceilingPolynomial p b+c^(5/2:ℝ))/c^2 = ceilingPolynomial p b+c^(1/2:ℝ) := by |
| rw [add_div,mul_div_cancel_left₀ _ (pow_ne_zero _ hc.ne'),rpow_five_halves_div_square hc] |
| unfold capIncrementQuotient |
| rw [← add_assoc] at hi |
| exact ⟨div_nonneg hi.1 (sq_nonneg c),(div_le_div_of_nonneg_right hi.2 (sq_nonneg c)).trans_eq he⟩ |
| |
| lemma capIncrementQuotient_inside {p c b : ℝ} (hp : 0 ≤ p) (hp2 : p ≤ 2) |
| (hc : 0 < c) (hb : |b| ≤ 1/2) : |
| capIncrementQuotient p c b = ceilingPolynomial p b+c^(1/2:ℝ) := by |
| have hd := ceiling_nonneg hp hp2 b |
| have hn : 0 ≤ c^2*ceilingPolynomial p b+c^(5/2:ℝ) := by positivity |
| unfold capIncrementQuotient |
| rw [max_eq_left (by linarith),max_eq_left (by linarith)] |
| have he : ((1/2:ℝ)+c^2*ceilingPolynomial p b+c^(5/2:ℝ)-|b|-(1/2-|b|)) = |
| c^2*ceilingPolynomial p b+c^(5/2:ℝ) := by ring |
| rw [he,add_div,mul_div_cancel_left₀ _ (pow_ne_zero _ hc.ne'),rpow_five_halves_div_square hc] |
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| lemma ceilingPolynomial_continuous (p : ℝ) : Continuous (ceilingPolynomial p) := by |
| unfold ceilingPolynomial |
| fun_prop |
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| lemma singular_ceiling_eventually (p : ℝ) : |
| ∀ᶠ c : ℝ in 𝓝[>] 0, ∀ b : ℝ, |b| ≤ 2 → |
| (1/2:ℝ)+c^2*ceilingPolynomial p b+c^(5/2:ℝ) ≤ 2 := by |
| obtain ⟨C,hC⟩ := (isCompact_Icc (a := (-2:ℝ)) (b := 2)).exists_bound_of_continuousOn |
| (ceilingPolynomial_continuous p).continuousOn |
| have ht : Tendsto (fun c : ℝ => c^2*C+c^(5/2:ℝ)) (𝓝[>] 0) (𝓝 0) := by |
| have h0 : Tendsto (fun c : ℝ => c) (𝓝[>] 0) (𝓝 0) := tendsto_id.mono_left nhdsWithin_le_nhds |
| simpa using ((h0.pow 2).mul_const C).add (right_rpow_tendsto_zero (q := (5/2:ℝ)) (by norm_num)) |
| filter_upwards [ht.eventually (gt_mem_nhds (by norm_num : (0:ℝ)<3/2))] with c hc b hb |
| have h := (le_abs_self (ceilingPolynomial p b)).trans (by simpa only [Real.norm_eq_abs] using hC b (abs_le.mp hb)) |
| have hh := mul_le_mul_of_nonneg_left h (sq_nonneg c) |
| linarith |
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| lemma capIncrementQuotient_tendsto {p : ℝ} (hp : 0 ≤ p) (hp2 : p ≤ 2) (b : ℝ) : |
| Tendsto (fun c : ℝ => capIncrementQuotient p c b) (𝓝[>] 0) |
| (𝓝 ((Icc (-(1/2:ℝ)) (1/2)).indicator (ceilingPolynomial p) b)) := by |
| by_cases hb : |b| ≤ 1/2 |
| · rw [Set.indicator_of_mem (show b ∈ Icc (-(1/2:ℝ)) (1/2) from abs_le.mp hb)] |
| have h : Tendsto (fun c : ℝ => ceilingPolynomial p b+c^(1/2:ℝ)) (𝓝[>] 0) |
| (𝓝 (ceilingPolynomial p b)) := by |
| simpa using (right_rpow_tendsto_zero (q := (1/2:ℝ)) (by norm_num)).const_add (ceilingPolynomial p b) |
| apply h.congr' |
| filter_upwards [self_mem_nhdsWithin] with c hc |
| exact (capIncrementQuotient_inside hp hp2 hc hb).symm |
| · rw [Set.indicator_of_notMem (by simpa only [mem_Icc,← abs_le] using hb)] |
| have ht : Tendsto (fun c : ℝ => (1/2:ℝ)+c^2*ceilingPolynomial p b+c^(5/2:ℝ)) |
| (𝓝[>] 0) (𝓝 (1/2:ℝ)) := by |
| have h0 : Tendsto (fun c : ℝ => c) (𝓝[>] 0) (𝓝 0) := tendsto_id.mono_left nhdsWithin_le_nhds |
| simpa using (tendsto_const_nhds.add ((h0.pow 2).mul_const (ceilingPolynomial p b))).add |
| (right_rpow_tendsto_zero (q := (5/2:ℝ)) (by norm_num)) |
| apply tendsto_const_nhds.congr' |
| filter_upwards [ht.eventually (gt_mem_nhds (lt_of_not_ge hb))] with c hc |
| unfold capIncrementQuotient |
| rw [max_eq_right (by linarith),max_eq_right (by linarith)] |
| simp |
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|
| lemma capIncrementQuotient_integral_tendsto {p : ℝ} (hp : 0 ≤ p) (hp2 : p ≤ 2) : |
| Tendsto (fun c : ℝ => ∫ b in Icc (-2:ℝ) 2, capIncrementQuotient p c b) (𝓝[>] 0) |
| (𝓝 (∫ b in Icc (-(1/2:ℝ)) (1/2), ceilingPolynomial p b)) := by |
| have hm : ∀ c : ℝ, Continuous (capIncrementQuotient p c) := by |
| intro c |
| unfold capIncrementQuotient |
| exact (((continuous_const.add (continuous_const.mul (ceilingPolynomial_continuous p))).add continuous_const).sub continuous_abs).max continuous_const |>.sub |
| ((continuous_const.sub continuous_abs).max continuous_const) |>.div_const _ |
| have h := tendsto_integral_filter_of_dominated_convergence |
| (μ := volume.restrict (Icc (-2:ℝ) 2)) (fun b => ceilingPolynomial p b+1) |
| (Filter.Eventually.of_forall (fun c => (hm c).aestronglyMeasurable)) |
| (show ∀ᶠ c : ℝ in 𝓝[>] 0, ∀ᵐ b ∂volume.restrict (Icc (-2:ℝ) 2), |
| ‖capIncrementQuotient p c b‖ ≤ ceilingPolynomial p b+1 from by |
| have he : ∀ᶠ c : ℝ in 𝓝[>] 0, c ≤ 1 := |
| (eventually_lt_nhds (by norm_num : (0:ℝ)<1)).filter_mono nhdsWithin_le_nhds |>.mono (fun _ h => h.le) |
| filter_upwards [self_mem_nhdsWithin,he] with c hc hc1 |
| exact Filter.Eventually.of_forall (fun b => by |
| obtain ⟨h0,h1⟩ := capIncrementQuotient_nonneg_bound hp hp2 hc (b := b) |
| rw [Real.norm_eq_abs,abs_of_nonneg h0] |
| exact h1.trans (by linarith [Real.rpow_le_one (z := (1/2:ℝ)) hc.le hc1 (by norm_num)]))) |
| (((ceilingPolynomial_continuous p).add continuous_const).continuousOn.integrableOn_Icc) |
| (Filter.Eventually.of_forall (capIncrementQuotient_tendsto hp hp2)) |
| have he : (∫ b in Icc (-2:ℝ) 2, (Icc (-(1/2:ℝ)) (1/2)).indicator (ceilingPolynomial p) b) = |
| ∫ b in Icc (-(1/2:ℝ)) (1/2), ceilingPolynomial p b := by |
| rw [integral_indicator measurableSet_Icc,Measure.restrict_restrict measurableSet_Icc] |
| congr 2 |
| apply Set.inter_eq_left.mpr |
| intro b hb |
| constructor <;> linarith [hb.1,hb.2] |
| exact he ▸ h |
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| lemma singularDomain_volume_difference {p gamma c : ℝ} (hp : 0 ≤ p) (hp2 : p ≤ 2) |
| (hc : 0 < c) (hd : c^gamma ≤ (1/2:ℝ)) |
| (hcap : ∀ b : ℝ, |b| ≤ 2 → (1/2:ℝ)+c^2*ceilingPolynomial p b+c^(5/2:ℝ) ≤ 2) : |
| (volume.real (singularDomain p gamma c)-(1/4-(c^gamma)^2))/c^2 = |
| ∫ b in Icc (-2:ℝ) 2, capIncrementQuotient p c b := by |
| have hv := coneBand_volume (d := c^gamma) (a := (1/2:ℝ)) (by positivity) hd |
| norm_num at hv |
| rw [← hv,singularDomain_volume_vertical p gamma c hcap,coneBand_volume_vertical (by norm_num)] |
| have hI₁ : IntegrableOn (fun b : ℝ => max ((1/2:ℝ)+c^2*ceilingPolynomial p b+c^(5/2:ℝ)-max (c^gamma) |b|) 0) |
| (Icc (-2:ℝ) 2) := by |
| apply ContinuousOn.integrableOn_Icc |
| apply Continuous.continuousOn |
| exact (((continuous_const.add (continuous_const.mul (ceilingPolynomial_continuous p))).add continuous_const).sub |
| (continuous_const.max continuous_abs)).max continuous_const |
| have hI₂ : IntegrableOn (fun b : ℝ => max ((1/2:ℝ)-max (c^gamma) |b|) 0) (Icc (-2:ℝ) 2) := by |
| apply ContinuousOn.integrableOn_Icc |
| exact ((continuous_const.sub (continuous_const.max continuous_abs)).max continuous_const).continuousOn |
| rw [← integral_sub hI₁ hI₂,← integral_div] |
| apply setIntegral_congr_fun measurableSet_Icc |
| intro b _ |
| unfold capIncrementQuotient |
| apply congrArg (fun x : ℝ => x/c^2) |
| have he := positivepart_floor_increment (a := (1/2:ℝ)) (d := c^gamma) |
| (h := c^2*ceilingPolynomial p b+c^(5/2:ℝ)) (t := |b|) hd |
| (by have := ceiling_nonneg hp hp2 b; positivity) |
| simpa only [← add_assoc] using he |
|
|
| lemma singularDomain_volume_tendsto {p gamma : ℝ} (hp : 0 ≤ p) (hp2 : p ≤ 2) (hg : 1 < gamma) : |
| Tendsto (fun c : ℝ => (volume.real (singularDomain p gamma c)-1/4)/c^2) (𝓝[>] 0) |
| (𝓝 (∫ b in Icc (-(1/2:ℝ)) (1/2), ceilingPolynomial p b)) := by |
| have h := (capIncrementQuotient_integral_tendsto hp hp2).sub (singular_tip_small hg).tendsto_div_nhds_zero |
| simp only [sub_zero] at h |
| apply h.congr' |
| filter_upwards [self_mem_nhdsWithin,singular_ceiling_eventually p, |
| (right_rpow_tendsto_zero (lt_trans (by norm_num) hg)).eventually (gt_mem_nhds (by norm_num : (0:ℝ)<1/2))] with c hc hcap hd |
| rw [← singularDomain_volume_difference hp hp2 hc hd.le hcap] |
| ring |
| |
| end RuledApproximation |
| |
| end |
| |
| end OAI |
| |
