| import OAI.Geometry.RuledCovering.EnergyIntegral |
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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 singular_weighted_integral_tendsto {p gamma : ℝ} (hp : 1 < p) (hp2 : p ≤ 2) (hg : 1 < gamma) : |
| Tendsto (fun c : ℝ => |
| ((∫ z in singularDomain p gamma c, physicalWeight (Real.sqrt 2) (c • singularField p z))-1/4)/c^2) |
| (𝓝[>] 0) (𝓝 (singularCoefficient p)) := by |
| have ht := (singularDomain_volume_tendsto (by linarith : 0 ≤ p) hp2 hg).add |
| (energyCorrection_integral_tendsto hp hp2 (by linarith : 0 < gamma)) |
| have hlim : (∫ b in Icc (-(1/2:ℝ)) (1/2), ceilingPolynomial p b) + |
| -(∫ z in coneTriangle (1/2), singularEnergy p z)/2 = singularCoefficient p := by |
| rw [neg_div,← sub_eq_add_neg] |
| exact singular_exact_coefficient hp |
| rw [hlim] at ht |
| apply ht.congr' |
| filter_upwards [self_mem_nhdsWithin] with c hc |
| have hc0 : c ≠ 0 := ne_of_gt hc |
| rw [singular_weighted_integral_expansion hp gamma c] |
| field_simp |
| ring |
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| lemma singular_totalCost_quotient_tendsto {p gamma : ℝ} (hp : 1 < p) (hp2 : p ≤ 2) (hg : 1 < gamma) : |
| Tendsto (fun c : ℝ => |
| (2*singularCost p gamma c+8*Real.sqrt 2*(c^gamma)^2-Real.sqrt 2/2)/c^2) |
| (𝓝[>] 0) (𝓝 (2*Real.sqrt 2*singularCoefficient p)) := by |
| have ht := ((singular_weighted_integral_tendsto hp hp2 hg).const_mul (2*Real.sqrt 2)).add |
| ((singular_tip_small hg).tendsto_div_nhds_zero.const_mul (8*Real.sqrt 2)) |
| simp only [mul_zero,add_zero] at ht |
| apply ht.congr' |
| filter_upwards with c |
| unfold singularCost |
| ring |
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| lemma singular_asymptotic {p gamma : ℝ} (hp : 1 < p) (hp2 : p ≤ 2) (hg : 1 < gamma) : |
| Asymptotics.IsLittleO (𝓝[>] (0:ℝ)) |
| (fun c => 2*singularCost p gamma c+8*Real.sqrt 2*(c^gamma)^2- |
| (Real.sqrt 2/2+2*Real.sqrt 2*singularCoefficient p*c^2)) |
| (fun c => c^2) := by |
| rw [Asymptotics.isLittleO_iff_tendsto'] |
| · have ht := (singular_totalCost_quotient_tendsto hp hp2 hg).sub_const (2*Real.sqrt 2*singularCoefficient p) |
| simp only [sub_self] at ht |
| apply ht.congr' |
| filter_upwards [self_mem_nhdsWithin] with c hc |
| have hc0 : c ≠ 0 := ne_of_gt hc |
| field_simp |
| ring |
| · filter_upwards [self_mem_nhdsWithin] with c hc |
| exact fun h => False.elim ((pow_ne_zero 2 (ne_of_gt hc)) h) |
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| lemma singular_totalCost_small {p gamma : ℝ} (hp : 1 < p) (hp2 : p ≤ 2) (hg : 1 < gamma) |
| (hk : singularCoefficient p < 0) : |
| ∀ᶠ c : ℝ in 𝓝[>] 0, |
| 2*singularCost p gamma c+8*Real.sqrt 2*(c^gamma)^2 < Real.sqrt 2/2 := by |
| have hk' : 2*Real.sqrt 2*singularCoefficient p < 0 := mul_neg_of_pos_of_neg |
| (mul_pos (by norm_num) (Real.sqrt_pos.mpr (by norm_num))) hk |
| filter_upwards [self_mem_nhdsWithin, |
| (singular_totalCost_quotient_tendsto hp hp2 hg).eventually (gt_mem_nhds hk')] with c hc hcost |
| exact sub_neg.mp (by simpa only [zero_mul] using (div_lt_iff₀ (sq_pos_of_pos hc)).mp hcost) |
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| end RuledApproximation |
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| end |
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| end OAI |
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