| import OAI.Geometry.LatticeCovering.GaussianRates |
|
|
| namespace OAI |
|
|
|
|
| namespace SingleLatticeCovering.FinalRates |
| open Real Filter Topology Asymptotics |
|
|
| noncomputable def localScale (C : ℝ) (n : ℕ) : ℕ := ⌈sectionRadius C n⌉₊+2 |
|
|
| lemma sectionRadius_le_localScale (C : ℝ) (n : ℕ) : |
| sectionRadius C n ≤ (localScale C n : ℝ) := by |
| have H := Nat.le_ceil (sectionRadius C n) |
| dsimp [localScale] |
| push_cast |
| linarith |
|
|
| lemma dimension_le_localScale {C : ℝ} (hC : 0 < C) {n D : ℕ} |
| (hD : (D:ℝ) ≤ C*((firstSize n:ℝ)+1)) : D ≤ localScale C n := by |
| have _ := hC |
| have hc := sectionRadius_le_localScale C n |
| have hh : (D:ℝ) ≤ (localScale C n:ℝ) := by |
| dsimp [sectionRadius] at hc |
| nlinarith [show (0:ℝ) ≤ (firstSize n:ℝ) by positivity] |
| exact_mod_cast hh |
|
|
| lemma localScale_tendsto {C : ℝ} (hC : 0 < C) : Tendsto (localScale C) atTop atTop := by |
| apply tendsto_atTop_mono _ firstSize_tendsto |
| intro n |
| have H := sectionRadius_le_localScale C n |
| dsimp [sectionRadius] at H |
| exact_mod_cast (show (firstSize n:ℝ) ≤ (localScale C n:ℝ) by |
| nlinarith [show (0:ℝ) ≤ (firstSize n:ℝ) by positivity]) |
|
|
| lemma eventually_localScale_root {C : ℝ} (hC : 0 < C) : ∀ᶠ n : ℕ in atTop, |
| Real.exp ((localScale C n:ℝ)^30) ≤ Real.sqrt (Real.sqrt (n:ℝ)) := by |
| let H : ℝ := 3*(C+2) |
| have hH : 0 < H := by dsimp [H]; positivity |
| have hsmall : (fun n : ℕ => (Real.log (Real.log (n:ℝ)))^60) =o[atTop] |
| (fun n => Real.log (n:ℝ)) := by |
| simpa only [Function.comp_def,Real.rpow_natCast,Real.rpow_one] using |
| (isLittleO_log_rpow_rpow_atTop ((60:ℕ):ℝ) (by norm_num : (0:ℝ) < 1)).comp_tendsto log_nat_atTop |
| filter_upwards [hsmall.bound (by positivity : 0 < 1/(4*H^30)), |
| loglog_nat_atTop.eventually (eventually_ge_atTop (1:ℝ)), |
| eventually_ge_atTop (1:ℕ)] with n hbound hs hn |
| have hnp : (0:ℝ) < n := by exact_mod_cast hn |
| have hs0 : 0 ≤ Real.log (Real.log (n:ℝ)) := le_trans zero_le_one hs |
| have hs2 : 1 ≤ (Real.log (Real.log (n:ℝ)))^2 := one_le_pow₀ hs |
| have hceil := Nat.ceil_lt_add_one (sq_nonneg (Real.log (Real.log (n:ℝ)))) |
| change (firstSize n:ℝ) < (Real.log (Real.log (n:ℝ)))^2+1 at hceil |
| have hlceil := Nat.ceil_lt_add_one (sectionRadius_pos hC n).le |
| have hl : (localScale C n:ℝ) ≤ H*(Real.log (Real.log (n:ℝ)))^2 := by |
| dsimp [localScale,sectionRadius] |
| push_cast |
| dsimp [sectionRadius] at hlceil |
| have hh := mul_le_mul_of_nonneg_left hceil.le (show 0 ≤ C+1 by linarith) |
| dsimp [H] |
| nlinarith [mul_nonneg (show 0 ≤ 2*C+5 by linarith) (sub_nonneg.mpr hs2)] |
| rw [Real.norm_eq_abs,Real.norm_eq_abs,abs_of_nonneg (pow_nonneg hs0 _), |
| abs_of_nonneg (Real.log_natCast_nonneg n)] at hbound |
| have hpow : (localScale C n:ℝ)^30 ≤ H^30*(Real.log (Real.log (n:ℝ)))^60 := by |
| have hh := pow_le_pow_left₀ (Nat.cast_nonneg (localScale C n)) hl 30 |
| simpa only [mul_pow,←pow_mul] using hh |
| have hexp : 4*(localScale C n:ℝ)^30 ≤ Real.log (n:ℝ) := by |
| have hh := mul_le_mul_of_nonneg_left hbound (by positivity : 0 ≤ 4*H^30) |
| have he : (4*H^30)*(1/(4*H^30)*Real.log (n:ℝ))=Real.log (n:ℝ) := by field_simp |
| rw [he] at hh |
| nlinarith |
| have hlog : Real.log (Real.sqrt (Real.sqrt (n:ℝ)))=Real.log (n:ℝ)/4 := by |
| rw [Real.log_sqrt (Real.sqrt_nonneg _),Real.log_sqrt hnp.le] |
| ring |
| rw [←Real.exp_log (Real.sqrt_pos.mpr (Real.sqrt_pos.mpr hnp))] |
| apply Real.exp_le_exp.mpr |
| rw [hlog] |
| linarith |
|
|
|
|
| end SingleLatticeCovering.FinalRates |
|
|
|
|
|
|
| namespace SingleLatticeCovering.GaussianNumerics |
| open MeasureTheory MeasureTheory.Measure Set Filter Topology GaussianDensity CanonicalDensity Sections |
| open scoped ENNReal |
|
|
| lemma normalizer_half (D : ℕ) : normalizer D (1/2)=(Real.sqrt (2*Real.pi))^D := by |
| unfold normalizer |
| rw [show Real.pi/(1/2)=2*Real.pi by ring,Real.sqrt_eq_rpow, |
| ←Real.rpow_natCast,←Real.rpow_mul (by positivity : 0 ≤ 2*Real.pi)] |
| congr 1 |
| ring |
|
|
| lemma density_half_gaussian {D : ℕ} (y : E D) : |
| GaussianDensity.density (1/2) y=Sections.gaussian (E D) y := by |
| unfold GaussianDensity.density Sections.gaussian |
| rw [normalizer_half,finrank_euclideanSpace_fin,zpow_neg,zpow_natCast] |
| congr 1 |
| congr 1 |
| ring |
|
|
|
|
| theorem eventually_local_coordinate_position : ∀ᶠ l : ℕ in atTop, |
| ∀ m D : ℕ, D ≤ l → 0 < m+D → |
| Real.exp ((l:ℝ)^30) ≤ Real.sqrt (Real.sqrt ((m+D : ℕ):ℝ)) → |
| ∀ K : Set (Fin (m+D) → ℝ), IsCompact K → Convex ℝ K → (interior K).Nonempty → |
| ∃ e : (Fin (m+D) → ℝ) ≃ᵃ[ℝ] (Fin (m+D) → ℝ), |
| ∀ y : Fin D → ℝ, ‖WithLp.toLp 2 y‖ ≤ 6*(l:ℝ)^2 → |
| (volume (e '' K)).toReal*Folded.gamma y/2 ≤ |
| (volume (coordinateFiber (e '' K) y)).toReal := by |
| filter_upwards [eventually_local_position] with l hl |
| intro m D hD hn hroot K hK hc hi |
| let A : (Fin (m+D) → ℝ) ≃ₗ[ℝ] E (m+D) := (WithLp.linearEquiv 2 ℝ (Fin (m+D) → ℝ)).symm |
| have hAK : IsCompact (A '' K) := hK.image A.toContinuousLinearEquiv.continuous |
| have hAc : Convex ℝ (A '' K) := hc.linear_image A.toLinearMap |
| have hAi : (interior (A '' K)).Nonempty := by |
| change (interior (A.toContinuousLinearEquiv.toHomeomorph '' K)).Nonempty |
| rw [←A.toContinuousLinearEquiv.toHomeomorph.image_interior] |
| exact hi.image _ |
| obtain ⟨g,hg⟩ := hl m D hD hn hroot (A '' K) hAK hAc hAi |
| let J := g '' (A '' K) |
| have hJ : IsCompact J := hAK.image g.toAffineMap.continuous_of_finiteDimensional |
| let G : E (m+D) ≃ₜ ((Fin m → ℝ) × E D) := |
| { toEquiv := g.toEquiv |
| continuous_toFun := g.toAffineMap.continuous_of_finiteDimensional |
| continuous_invFun := g.symm.toAffineMap.continuous_of_finiteDimensional } |
| have hJi : (interior J).Nonempty := by |
| change (interior (G '' (A '' K))).Nonempty |
| rw [←G.image_interior] |
| exact hAi.image G |
| have hJ0 : 0 < (volume J).toReal := by |
| apply ENNReal.toReal_pos (ne_of_gt ?_) hJ.measure_ne_top |
| exact (isOpen_interior.measure_pos volume hJi).trans_le (measure_mono interior_subset) |
| let e := A.toAffineEquiv.trans (g.trans (flatten m D).toAffineEquiv) |
| have he : e '' K=flatten m D '' J := by |
| dsimp [e,J] |
| rw [Set.image_image,Set.image_image] |
| rfl |
| refine ⟨e,fun y hy => ?_⟩ |
| rw [he,flatten_volume m D hJ] |
| have hf : coordinateFiber (flatten m D '' J) y=Sections.fiber J (WithLp.toLp 2 y) := by |
| have H := flatten_fiber (flatten m D '' J) (WithLp.toLp 2 y) |
| have hc : (flatten m D).symm '' ((flatten m D) '' J)=J := by |
| simp only [Set.image_image,LinearEquiv.symm_apply_apply,Set.image_id'] |
| rw [hc] at H |
| exact H.symm |
| rw [hf] |
| have H := hg (WithLp.toLp 2 y) hy |
| rw [density_half_gaussian,gaussian_toLp] at H |
| change Folded.gamma y/2 ≤ canonical J (WithLp.toLp 2 y) at H |
| rw [canonical,ENNReal.toReal_inv] at H |
| have HH := mul_le_mul_of_nonneg_left H hJ0.le |
| rw [←mul_assoc,mul_inv_cancel₀ hJ0.ne',one_mul] at HH |
| simpa only [mul_div_assoc] using HH |
|
|
|
|
| end SingleLatticeCovering.GaussianNumerics |
|
|
|
|
|
|
|
|
| noncomputable section |
|
|
| end |
|
|
| end OAI |
|
|
