import OAI.Geometry.LatticeCovering.Rates namespace OAI section section noncomputable section namespace SingleLatticeCovering.Horizontal open MeasureTheory Completion open scoped ENNReal BigOperators structure FullLattice (m : ℕ) where module : Submodule ℤ (Fin m → ℝ) discrete : DiscreteTopology module full : IsZLattice ℝ module attribute [instance] FullLattice.discrete FullLattice.full namespace FullLattice def hole {m : ℕ} (L : FullLattice m) (J : Set (Fin m → ℝ)) : ℝ≥0∞ := torusMeasure m ((latticeProjection L.module '' J)ᶜ) end FullLattice theorem common_horizontal_from_mean {m : ℕ} {X α : Type*} [MeasurableSpace X] (μ : Measure X) [IsProbabilityMeasure μ] (family : X → FullLattice m) (J : α → Set (Fin m → ℝ)) (s : Finset α) (Dh CR η τ N : ℝ) (hCR : 0 ≤ CR) (hDh : 0 < Dh) (hcov : ∀ x, ZLattice.covolume (family x).module=Dh) (hmeas : ∀ a ∈ s, Measurable (fun x => (family x).hole (J a))) (hmean : ∀ a ∈ s, ∫⁻ x, (family x).hole (J a) ∂μ ≤ ENNReal.ofReal (Real.exp (-(volume (J a)).toReal/Dh)+CR*Real.exp (-η))) (hlo : ∀ a ∈ s, τ ≤ (volume (J a)).toReal/Dh) (hhi : ∀ a ∈ s, (volume (J a)).toReal/Dh ≤ η) (hcard : (s.card : ℝ) ≤ N) (hcriterion : (1+CR)*N*Real.exp (-τ/2)<1) : ∃ L : FullLattice m, ZLattice.covolume L.module=Dh ∧ ∀ a ∈ s, L.hole (J a) ≤ ENNReal.ofReal (Real.exp (-((volume (J a)).toReal/Dh)/2)) := by have _ := hDh have hsum : (1+CR)*(∑ a ∈ s, Real.exp (-((volume (J a)).toReal/Dh)/2))<1 := by apply lt_of_le_of_lt _ hcriterion calc _ ≤ (1+CR)*(∑ a ∈ s, Real.exp (-τ/2)) := mul_le_mul_of_nonneg_left (Finset.sum_le_sum (fun a ha => Real.exp_le_exp.mpr (by linarith [hlo a ha]))) (by linarith) _ = (1+CR)*(s.card : ℝ)*Real.exp (-τ/2) := by rw [Finset.sum_const, nsmul_eq_mul]; ring _ ≤ _ := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hcard (by linarith)) (Real.exp_pos _).le obtain ⟨x,hx⟩ := common_hole_sample μ s (fun a x => (family x).hole (J a)) (fun a => (volume (J a)).toReal/Dh) CR η hCR hmeas hhi (by intro a ha simpa only [neg_div] using hmean a ha) hsum exact ⟨family x,hcov x,hx⟩ end SingleLatticeCovering.Horizontal end end section noncomputable section namespace SingleLatticeCovering.Assembly open MeasureTheory Folded LatticeGeometry Completion Shear CoverGeometry Vertical Sections Horizontal open scoped BigOperators Pointwise ENNReal theorem sections_and_mean_cover {B : Block} (c : Chain B) {m : ℕ} (hm : 2 ≤ m) (hD : 0 < c.fullDim) (K : Set (Fin (m+c.fullDim) → ℝ)) (hK : IsCompact K) (hconv : Convex ℝ K) (hV : 0 < (volume K).toReal) {R : ℝ} (hR : 0 < R) (hrad : c.radiusSq+(B.terminalDim : ℝ) ≤ R^2) (hsmall : R^2/(m : ℝ)^2 ≤ 1) (hgauss : ∀ y : c.FullVec, ‖WithLp.toLp 2 y‖ ≤ 6*(c.fullDim : ℝ)*R → (volume K).toReal*gamma y/2 ≤ (volume (coordinateFiber K y)).toReal) (κ ρ ε τ A CR : ℝ) (hρ : 0 < ρ) (hτ : 0 ≤ τ) (hA : 0 < A) (hCR : 0 ≤ CR) (hpat : ∀ y : c.FullVec, ∃ P : Finset c.FullVec, P.Nonempty ∧ SuffixBinary P ∧ (∀ l ∈ P, l ∈ c.fullRaw ∧ c.FullResidual y l ∧ c.det*gamma (y-c.fullLinear l) ≤ ε) ∧ κ ≤ ∑ l ∈ P, c.det*gamma (y-c.fullLinear l)) (hload : A+(2^c.fullDim : ℕ)*τ ≤ (Real.exp (-1)/8)*κ*ρ) (hcap : (Real.exp 1/8)*ρ*ε ≤ FinalRates.eta m) (hcriterion : (1+CR)*(3*(m : ℝ)^2)^c.fullDim*Real.exp (-τ/2)<1) (hcount : ∀ {α : Type} (s : Finset α), (s.card : ℝ) ≤ (3*(m : ℝ)^2)^c.fullDim → let Lstar := 2*((patternTotal c.alphabetSeq s c.fullDim : ℝ)+1) 1 ≤ Lstar ∧ (patternTotal c.alphabetSeq s c.fullDim : ℝ) (family x).hole J)) (hmean : ∀ J : Set (Fin m → ℝ), IsCompact J → Convex ℝ J → (interior J).Nonempty → (volume J).toReal/((volume K).toReal/(ρ*c.det)) ≤ FinalRates.eta m → ∫⁻ x, (family x).hole J ∂μ ≤ ENNReal.ofReal (Real.exp (-(volume J).toReal/((volume K).toReal/(ρ*c.det)))+ CR*Real.exp (-FinalRates.eta m))) : ∃ (Λ : Submodule ℤ (Fin (m+c.fullDim) → ℝ)) (_ : DiscreteTopology Λ), IsZLattice ℝ Λ ∧ K+(Λ : Set (Fin (m+c.fullDim) → ℝ))=Set.univ ∧ (volume K).toReal/ZLattice.covolume Λ volume ≤ Real.exp 1*ρ := by classical let Dh := (volume K).toReal/(ρ*c.det) have hDh : 0 < Dh := div_pos hV (mul_pos hρ c.det_pos) obtain ⟨e,heK,heconv,hevol,S,hSc,J,hJ,hsections⟩ := coordinate_marginal_to_labels hm hD hK hconv hV hR hsmall hgauss let α := {a // a ∈ S} let J' : α → Set (Fin m → ℝ) := fun a => J a.val let u : α → ℝ := fun a => (volume (J' a)).toReal/Dh let s : Finset α := Finset.univ.filter (fun a => τ ≤ u a ∧ u a ≤ (Real.exp 1/8)*ρ*ε) have hcard : (s.card : ℝ) ≤ (3*(m : ℝ)^2)^c.fullDim := by have hh : s.card ≤ S.card := by calc s.card ≤ (Finset.univ : Finset α).card := Finset.card_le_card (Finset.filter_subset _ _) _ = S.card := by simp [α] exact (by exact_mod_cast hh : (s.card : ℝ) ≤ S.card).trans hSc have hulo (a : α) (ha : a ∈ s) : τ ≤ u a := (Finset.mem_filter.mp ha).2.1 have huhi (a : α) (ha : a ∈ s) : u a ≤ FinalRates.eta m := ((Finset.mem_filter.mp ha).2.2).trans hcap obtain ⟨L,hLc,hLhole⟩ := common_horizontal_from_mean μ family J' s Dh CR (FinalRates.eta m) τ ((3*(m : ℝ)^2)^c.fullDim) hCR hDh hcov (by intro a ha exact hmeas (J' a) (hJ a.val a.property).1 (hJ a.val a.property).2.1 (hJ a.val a.property).2.2) (by intro a ha exact hmean (J' a) (hJ a.val a.property).1 (hJ a.val a.property).2.1 (hJ a.val a.property).2.2 (huhi a ha)) hulo huhi hcard hcriterion let Lstar := 2*((patternTotal c.alphabetSeq s c.fullDim : ℝ)+1) obtain ⟨hLstar,hcost,herror⟩ := hcount s hcard have hV' : 0 < (volume (e '' K)).toReal := by rwa [hevol] have hcov' : ZLattice.covolume L.module volume=(volume (e '' K)).toReal/(ρ*c.det) := by rw [hevol]; exact hLc obtain ⟨Λ,hd,hfull,hcover,hbound⟩ := labeled_chain_to_cover c (by omega) L.module (e '' K) heK heconv hV' J' (fun a => (hJ a.val a.property).1) κ ρ (Real.exp (-1)/8) (Real.exp 1/8) ε τ A Lstar hρ (by positivity) (by positivity) hτ hA hcov' hpat (by intro v hv obtain ⟨a,ha,hin,hl,hu⟩ := hsections v (hv.trans hrad) exact ⟨⟨a,ha⟩,hin,hl,hu⟩) (by intro a hlo hhi have ha : a ∈ s := by apply Finset.mem_filter.mpr refine ⟨Finset.mem_univ _,?_,?_⟩ · simpa only [hLc] using hlo · simpa only [hLc] using hhi simpa only [FullLattice.hole,hLc] using hLhole a ha) hload hLstar (by simpa only [hLc,s,u,patternTotal,Lstar] using hcost) (by simpa only [Nat.cast_pow,Nat.cast_ofNat] using herror) let := hd let := hfull exact pull_back_cover e K Λ hcover hbound end SingleLatticeCovering.Assembly end end section noncomputable section namespace SingleLatticeCovering.Inputs open MeasureTheory Completion Horizontal Sections open scoped ENNReal structure MeanHoleModel (m : ℕ) (Dh CR : ℝ) where Sample : Type measurable : MeasurableSpace Sample measure : @Measure Sample measurable probability : @IsProbabilityMeasure Sample measurable measure lattice : Sample → FullLattice m covolume : ∀ x, ZLattice.covolume (lattice x).module volume=Dh measurable_hole : letI := measurable ∀ J : Set (Fin m → ℝ), IsCompact J → Convex ℝ J → (interior J).Nonempty → Measurable (fun x => (lattice x).hole J) mean : letI := measurable ∀ J : Set (Fin m → ℝ), IsCompact J → Convex ℝ J → (interior J).Nonempty → (volume J).toReal/Dh ≤ FinalRates.eta m → ∫⁻ x, (lattice x).hole J ∂measure ≤ ENNReal.ofReal (Real.exp (-(volume J).toReal/Dh)+CR*Real.exp (-FinalRates.eta m)) def GaussianPositions (α β γ C₀ : ℝ) (n : ℕ) : Prop := ∀ m D : ℕ, m+D=n → 0 < D → (D : ℝ) ≤ (n : ℝ)^α → ∀ 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‖ ≤ (n : ℝ)^β → (1-C₀*(n : ℝ)^(-γ))*(volume (e '' K)).toReal*Folded.gamma y ≤ (volume (coordinateFiber (e '' K) y)).toReal end SingleLatticeCovering.Inputs end end section noncomputable section namespace SingleLatticeCovering.FinalRates open Real Filter Topology def sectionRadius (C : ℝ) (n : ℕ) : ℝ := (C+1)*((firstSize n : ℝ)+1) lemma sectionRadius_pos {C : ℝ} (hC : 0 < C) (n : ℕ) : 0 < sectionRadius C n := by dsimp [sectionRadius]; positivity lemma radius_sq_bound {C b r : ℝ} (hC : 0 ≤ C) (hb : 0 ≤ b) (hr : r ≤ C*(b*Real.log (b+1)+1)) : r ≤ ((C+1)*(b+1))^2 := by have hl : Real.log (b+1) ≤ b := by simpa using Real.log_le_sub_one_of_pos (by linarith : 0 < b+1) have hinner : b*Real.log (b+1)+1 ≤ (b+1)^2 := by nlinarith [mul_le_mul_of_nonneg_left hl hb] calc r ≤ C*(b*Real.log (b+1)+1) := hr _ ≤ C*(b+1)^2 := mul_le_mul_of_nonneg_left hinner hC _ ≤ (C+1)^2*(b+1)^2 := mul_le_mul_of_nonneg_right (by nlinarith [sq_nonneg C]) (sq_nonneg _) _ = _ := by ring lemma eventually_first_polynomial {H r ε : ℝ} (hH : 0 < H) (hr : 0 < r) (hε : 0 < ε) (k : ℕ) : ∀ᶠ n : ℕ in atTop, H*((firstSize n : ℝ)+1)^k ≤ ε*(n : ℝ)^r := by filter_upwards [eventually_firstSize_le_log (by norm_num : (0 : ℝ) < 1), eventually_log_power_bound k hr (div_pos hε hH)] with n hfirst hlog simp only [one_mul] at hfirst calc _ ≤ H*(Real.log (n : ℝ))^k := mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (by positivity) hfirst k) hH.le _ ≤ H*((ε/H)*(n : ℝ)^r) := mul_le_mul_of_nonneg_left hlog hH.le _ = _ := by field_simp lemma eventually_marginal_scales {C α β γ C₀ : ℝ} (hC : 0 < C) (hα : 0 < α) (hβ : 0 < β) (hγ : 0 < γ) (hC₀ : 0 ≤ C₀) : ∀ᶠ n : ℕ in atTop, ∀ D : ℕ, (D : ℝ) ≤ C*((firstSize n : ℝ)+1) → D ≤ n ∧ 2 ≤ n-D ∧ (D : ℝ) ≤ (n : ℝ)^α ∧ 6*(D : ℝ)*sectionRadius C n < (n : ℝ)^β ∧ C₀*(n : ℝ)^(-γ) ≤ 1/2 ∧ sectionRadius C n^2/((n-D : ℕ) : ℝ)^2 ≤ 1 := by have _ := hC₀ have herr : ∀ᶠ n : ℕ in atTop, C₀*(n : ℝ)^(-γ) ≤ 1/2 := by have ht : Tendsto (fun n : ℕ => C₀*(n : ℝ)^(-γ)) atTop (𝓝 0) := by have hn : Tendsto (fun n : ℕ => (n : ℝ)) atTop atTop := tendsto_natCast_atTop_atTop simpa only [mul_zero,Function.comp_def] using ((tendsto_rpow_neg_atTop hγ).comp hn).const_mul C₀ exact ht.eventually (eventually_le_nhds (by norm_num : (0 : ℝ) < 1/2)) filter_upwards [eventually_first_polynomial hC hα (by norm_num : (0 : ℝ) < 1) 1, eventually_first_polynomial (show 0 < 6*C*(C+1) by positivity) hβ (by norm_num : (0 : ℝ) < 1/2) 2, eventually_first_polynomial (show 0 < C+1 by linarith) (by norm_num : (0 : ℝ) < 1) (by norm_num : (0 : ℝ) < 1/2) 1, eventually_dimension_small hC,herr,eventually_ge_atTop (4 : ℕ)] with n hDim hRad hSmall hDsmall he hn D hD simp only [pow_one,one_mul] at hDim simp only [pow_one,Real.rpow_one] at hSmall have hn4 : (4 : ℝ) ≤ n := by exact_mod_cast hn have hn0 : (0 : ℝ) < n := by linarith have hdhalf := (hDsmall D hD).2 have hDn : D ≤ n := by exact_mod_cast (show (D : ℝ) ≤ n by linarith) have hmhalf : (n : ℝ)/2 ≤ ((n-D : ℕ) : ℝ) := by rw [Nat.cast_sub hDn]; linarith have hm2 : (2 : ℝ) ≤ ((n-D : ℕ) : ℝ) := by linarith refine ⟨hDn,by exact_mod_cast hm2,hD.trans hDim,?_,he,?_⟩ · have hDR : 6*(D : ℝ)*sectionRadius C n ≤ 6*C*(C+1)*((firstSize n : ℝ)+1)^2 := by dsimp [sectionRadius] have hh := mul_le_mul_of_nonneg_right hD (show 0 ≤ 6*((C+1)*((firstSize n : ℝ)+1)) by positivity) nlinarith have hp : 0 < (n : ℝ)^β := Real.rpow_pos_of_pos hn0 β exact hDR.trans_lt (hRad.trans_lt (by nlinarith)) · have hRm : sectionRadius C n ≤ ((n-D : ℕ) : ℝ) := by dsimp [sectionRadius] linarith apply (div_le_one (sq_pos_of_pos (by linarith : (0 : ℝ) < ((n-D : ℕ) : ℝ)))).mpr exact pow_le_pow_left₀ (sectionRadius_pos hC n).le hRm 2 end SingleLatticeCovering.FinalRates end end section noncomputable section namespace SingleLatticeCovering.Assembly open MeasureTheory Folded LatticeGeometry Completion Shear CoverGeometry Vertical Sections Horizontal Inputs open Filter Topology open scoped BigOperators Pointwise ENNReal lemma body_volume_pos {n : ℕ} {K : Set (Fin n → ℝ)} (hK : IsCompact K) (hi : (interior K).Nonempty) : 0 < (volume K).toReal := by apply ENNReal.toReal_pos (ne_of_gt ?_) hK.measure_ne_top exact (isOpen_interior.measure_pos volume hi).trans_le (measure_mono interior_subset) lemma affine_body {n : ℕ} (e : (Fin n → ℝ) ≃ᵃ[ℝ] (Fin n → ℝ)) {K : Set (Fin n → ℝ)} (hK : IsCompact K) (hc : Convex ℝ K) (hi : (interior K).Nonempty) : IsCompact (e '' K) ∧ Convex ℝ (e '' K) ∧ (interior (e '' K)).Nonempty := by let he : (Fin n → ℝ) ≃ₜ (Fin n → ℝ) := { toEquiv := e.toEquiv continuous_toFun := e.toAffineMap.continuous_of_finiteDimensional continuous_invFun := e.symm.toAffineMap.continuous_of_finiteDimensional } refine ⟨hK.image he.continuous, hc.affine_image e.toAffineMap,?_⟩ change (interior (he '' K)).Nonempty rw [←he.image_interior] exact hi.image he lemma chain_first_le_dim {B : Block} (c : Chain B) : c.first.b ≤ c.dim := by induction c with | base B => rfl | append c B w ih => exact ih.trans (Nat.le_add_right _ _) def coversAtBound (n : ℕ) (d : ℝ) : Prop := ∀ K : Set (Fin n → ℝ), IsCompact K → Convex ℝ K → (interior K).Nonempty → ∃ (Λ : Submodule ℤ (Fin n → ℝ)) (_ : DiscreteTopology Λ), IsZLattice ℝ Λ ∧ K+(Λ : Set (Fin n → ℝ))=Set.univ ∧ (volume K).toReal/ZLattice.covolume Λ volume ≤ d theorem eventual_main_reduction {α β γ C₀ CR : ℝ} (hα : 0 < α) (hβ : 0 < β) (hγ : 0 < γ) (hC₀ : 0 ≤ C₀) (hCR : 0 ≤ CR) (hGaussian : ∀ᶠ n : ℕ in atTop, GaussianPositions α β γ C₀ n) (hMean : ∀ m : ℕ, 2 ≤ m → ∀ Dh : ℝ, 0 < Dh → Nonempty (MeanHoleModel m Dh CR)) : ∃ C : ℝ, 0 < C ∧ ∀ᶠ n : ℕ in atTop, ∀ K : Set (Fin n → ℝ), IsCompact K → Convex ℝ K → (interior K).Nonempty → ∃ (Λ : Submodule ℤ (Fin n → ℝ)) (_ : DiscreteTopology Λ), IsZLattice ℝ Λ ∧ K+(Λ : Set (Fin n → ℝ))=Set.univ ∧ (volume K).toReal/ZLattice.covolume Λ volume ≤ C*(n : ℝ)*Real.log (n : ℝ) := by classical obtain ⟨cutoff,hcut,C,k,κ,hC,hk,hκ,hvertical⟩ := vertical_patterns_alphabets obtain ⟨Cstar,hstar,hconditions⟩ := FinalRates.final_conditions hC hk (show 0 < Real.exp (-1)/8 by positivity) (show 0 ≤ Real.exp 1/8 by positivity) hκ (show (1 : ℝ) ≤ 3 by norm_num) hCR refine ⟨Real.exp 1*Cstar,by positivity,?_⟩ filter_upwards [hconditions,FinalRates.eventually_marginal_scales hC hα hβ hγ hC₀, FinalRates.firstSize_tendsto.eventually (eventually_ge_atTop cutoff), hGaussian] with n hn hscale hfirst hgaussian obtain ⟨B,c,hcb,hcs,hDim,hRad,hAlphabet,hPattern⟩ := hvertical (FinalRates.firstSize n) hfirst have hDdim : (c.fullDim : ℝ) ≤ C*((FinalRates.firstSize n : ℝ)+1) := by simpa [Chain.fullDim,Nat.cast_add] using hDim obtain ⟨hDn,hm,hcap,hload,hcrit,hcount⟩ := hn.2 c.fullDim hDdim obtain ⟨_,_,hDpower,hRadius,hErr,hSmall⟩ := hscale c.fullDim hDdim have hD : 0 < c.fullDim := by have hfirstdim := chain_first_le_dim c rw [hcb] at hfirstdim dsimp [Chain.fullDim] omega let m := n-c.fullDim have hmn : m+c.fullDim=n := Nat.sub_add_cancel hDn have hmnle : m ≤ n := Nat.sub_le _ _ have hnpos : (0 : ℝ) < n := by exact_mod_cast (show 0 < n by omega) have hnlog : 0 < Real.log (n : ℝ) := Real.log_pos (by exact_mod_cast (show 1 < n by omega)) let ρ := Cstar*(n : ℝ)*Real.log (n : ℝ) let ε := Real.exp (-k*(FinalRates.firstSize n : ℝ)^(70/100 : ℝ)) let R := FinalRates.sectionRadius C n have hρ : 0 < ρ := by dsimp [ρ]; positivity have hrad : c.radiusSq+(B.terminalDim : ℝ) ≤ R^2 := FinalRates.radius_sq_bound hC.le (Nat.cast_nonneg _) hRad have step : coversAtBound (m+c.fullDim) ((Real.exp 1*Cstar)*(n : ℝ)*Real.log (n : ℝ)) := by intro K hK hc hi obtain ⟨e,he⟩ := hgaussian m c.fullDim hmn hD hDpower K hK hc hi obtain ⟨heK,hec,hei⟩ := affine_body e hK hc hi have heV := body_volume_pos heK hei have hg : ∀ y : c.FullVec, ‖WithLp.toLp 2 y‖ ≤ 6*(c.fullDim : ℝ)*R → (volume (e '' K)).toReal*gamma y/2 ≤ (volume (coordinateFiber (e '' K) y)).toReal := by intro y hy apply le_trans _ (he y (hy.trans hRadius.le)) have hy0 : 0 < gamma y := Finset.prod_pos (fun j _ => gamma1_pos _) nlinarith [mul_nonneg (show 0 ≤ 1-C₀*(n : ℝ)^(-γ)-1/2 by linarith) (mul_nonneg heV.le hy0.le)] obtain ⟨model⟩ := hMean m hm ((volume (e '' K)).toReal/(ρ*c.det)) (div_pos heV (mul_pos hρ c.det_pos)) let := model.measurable let := model.probability obtain ⟨Λ,hd,hfull,hcover,hbound⟩ := sections_and_mean_cover c hm hD (e '' K) heK hec heV (FinalRates.sectionRadius_pos hC n) hrad hSmall hg κ ρ ε (Real.sqrt (m : ℝ)) (8*(n : ℝ)*Real.log (n : ℝ)) CR hρ (Real.sqrt_nonneg _) (by positivity) hCR (by intro y obtain ⟨P,hPn,hPb,hPl,hPt,hPw⟩ := hPattern y exact ⟨P,hPn,hPb,hPl,hPw⟩) (by simpa only [Nat.cast_pow,Nat.cast_ofNat] using hload) hcap hcrit (by intro δ s hs have hs' : (s.card : ℝ) ≤ (3*(n : ℝ)^2)^c.fullDim := hs.trans (by gcongr) have ha : ∀ j < c.fullDim, Real.log ((c.alphabetSeq j).card+1 : ℝ) ≤ Real.log 8+(c.fullDim : ℝ)*Real.log ((c.fullDim : ℝ)+2)+(FinalRates.firstSize n : ℝ)^2 := by intro j hj simpa only [c.alphabetSeq_fin ⟨j,hj⟩] using hAlphabet ⟨j,hj⟩ simpa only [Nat.sub_add_cancel hDn] using hcount s c.alphabetSeq hs' ha) model.measure model.lattice model.covolume model.measurable_hole model.mean let := hd let := hfull have hb : (volume (e '' K)).toReal/ZLattice.covolume Λ volume ≤ (Real.exp 1*Cstar)*(n : ℝ)*Real.log (n : ℝ) := by convert hbound using 1 ; dsimp [ρ] ; ring exact pull_back_cover e K Λ hcover hb change coversAtBound n ((Real.exp 1*Cstar)*(n : ℝ)*Real.log (n : ℝ)) simpa only [hmn] using step end SingleLatticeCovering.Assembly end end section noncomputable section open Set Module MeasureTheory open scoped BigOperators Pointwise namespace SingleLatticeCovering.FiniteDimensions theorem maximal_basis {n : ℕ} (K : Set (Fin n → ℝ)) (hK : IsCompact K) (h0 : (0 : Fin n → ℝ) ∈ interior K) : ∃ b : Basis (Fin n) ℝ (Fin n → ℝ), (∀ j, b j ∈ K) ∧ ∀ x ∈ K, ∀ j, |b.repr x j| ≤ 1 := by classical obtain ⟨r,hr,hball⟩ := Metric.mem_nhds_iff.mp (mem_interior_iff_mem_nhds.mp h0) let v₀ : Fin n → (Fin n → ℝ) := fun j => (r/2) • Pi.single j (1 : ℝ) have hv₀ : ∀ j, v₀ j ∈ K := by intro j apply hball rw [Metric.mem_ball,dist_zero_right] simp only [v₀,norm_smul,Pi.norm_single,Real.norm_eq_abs,abs_one,mul_one, abs_of_pos (by positivity : 0 < r/2)] linarith have hdet₀ : (Matrix.of v₀).det = (r/2)^n := by have he : Matrix.of v₀ = (r/2) • (1 : Matrix (Fin n) (Fin n) ℝ) := by ext i j simp [v₀,Pi.single_apply,Matrix.one_apply,Pi.smul_apply,eq_comm] rw [he,Matrix.det_smul,Matrix.det_one,mul_one,Fintype.card_fin] have hS : IsCompact {v : Fin n → (Fin n → ℝ) | ∀ j, v j ∈ K} := isCompact_pi_infinite (fun _ => hK) have hcont : Continuous (fun v : Fin n → (Fin n → ℝ) => |(Matrix.of v).det|) := by fun_prop obtain ⟨v,hv,hmax⟩ := hS.exists_isMaxOn ⟨v₀,hv₀⟩ hcont.continuousOn have hdet : (Matrix.of v).det ≠ 0 := by have h : |(Matrix.of v₀).det| ≤ |(Matrix.of v).det| := hmax hv₀ rw [hdet₀,abs_of_pos (pow_pos (by positivity : 0 < r/2) _)] at h intro hz rw [hz,abs_zero] at h exact (not_le_of_gt (pow_pos (by positivity : 0 < r/2) _)) h let b := basisOfLinearIndependentOfCardEqFinrank' v (Matrix.linearIndependent_rows_of_det_ne_zero hdet) (by simp) have hb : (b : Fin n → (Fin n → ℝ)) = v := by simp [b] refine ⟨b,?_,?_⟩ · intro j rw [show b j=v j from congrFun hb j] exact hv j · intro x hx j have hupdate : (Function.update v j x) ∈ {v : Fin n → (Fin n → ℝ) | ∀ j, v j ∈ K} := by intro k by_cases hk : k=j · subst k; simpa using hx · simpa [hk] using hv k have hbound := hmax hupdate have hd : (Matrix.of (Function.update v j x)).det = b.repr x j*(Matrix.of v).det := by have hs := b.sum_repr x have ht := Matrix.det_updateRow_sum (Matrix.of b) j (fun k => b.repr x k) change (Matrix.of (Function.update (fun k => b k) j (∑ k, b.repr x k • b k))).det = b.repr x j*(Matrix.of b).det at ht rw [hs,hb] at ht exact ht change |(Matrix.of (Function.update v j x)).det| ≤ |(Matrix.of v).det| at hbound rw [hd,abs_mul] at hbound exact (mul_le_mul_iff_left₀ (abs_pos.mpr hdet)).mp (by simpa [mul_comm] using hbound) end SingleLatticeCovering.FiniteDimensions namespace SingleLatticeCovering.FiniteDimensions open MeasureTheory Module Set Completion Vertical LatticeGeometry open scoped BigOperators Pointwise lemma convex_basis_cube {n : ℕ} (hn : 0 < n) (K : Set (Fin n → ℝ)) (hconv : Convex ℝ K) (h0 : (0 : Fin n → ℝ) ∈ K) (b : Basis (Fin n) ℝ (Fin n → ℝ)) (hb : ∀ j, b j ∈ K) : Icc (0 : Fin n → ℝ) (fun _ => 1/(n : ℝ)) ⊆ b.equivFun.symm ⁻¹' K := by intro x hx have hnr : (0 : ℝ) < n := by exact_mod_cast hn have hsum : ∑ j, x j ≤ 1 := calc ∑ j, x j ≤ ∑ _j : Fin n, 1/(n : ℝ) := Finset.sum_le_sum (fun j _ => hx.2 j) _ = 1 := by simp [hnr.ne'] let w : Fin (n+1) → ℝ := Fin.cons (1-∑ j, x j) x let z : Fin (n+1) → (Fin n → ℝ) := Fin.cons 0 b have hw : ∀ j ∈ (Finset.univ : Finset (Fin (n+1))), 0 ≤ w j := by intro j _ refine Fin.cases ?_ (fun i => ?_) j · exact sub_nonneg.mpr hsum · exact hx.1 i have hw1 : ∑ j, w j = 1 := by simp [w,Fin.sum_univ_succ] have hz : ∀ j ∈ (Finset.univ : Finset (Fin (n+1))), z j ∈ K := by intro j _ exact Fin.cases h0 hb j have hc := hconv.sum_mem hw hw1 hz change b.equivFun.symm x ∈ K rw [Basis.equivFun_symm_apply] simpa only [Fin.sum_univ_succ,w,z,Fin.cons_zero,Fin.cons_succ,smul_zero,zero_add] using hc lemma integer_cube_cover {n : ℕ} (hn : 0 < n) : Icc (0 : Fin n → ℝ) (fun _ => 1/(n : ℝ))+ (contractLattice (integerLattice n) (n : ℝ) (by exact_mod_cast hn.ne') : Set (Fin n → ℝ)) = Set.univ := by apply Set.eq_univ_of_forall intro x have hnr : (0 : ℝ) < n := by exact_mod_cast hn let l : Fin n → ℝ := fun j => (⌊(n : ℝ)*x j⌋ : ℤ)/(n : ℝ) let q : Fin n → ℝ := fun j => Int.fract ((n : ℝ)*x j)/(n : ℝ) have hl : l ∈ contractLattice (integerLattice n) (n : ℝ) (by exact_mod_cast hn.ne') := by rw [mem_contractLattice] convert integer_mem n (fun j => ⌊(n : ℝ)*x j⌋) using 1 ext j simp only [Pi.smul_apply,smul_eq_mul,l] field_simp refine ⟨q,⟨?_,?_⟩,l,hl,?_⟩ · intro j exact div_nonneg (Int.fract_nonneg _) hnr.le · intro j exact (div_le_div_iff_of_pos_right hnr).mpr (Int.fract_lt_one _).le · ext j change Int.fract ((n : ℝ)*x j)/(n : ℝ)+(⌊(n : ℝ)*x j⌋ : ℤ)/(n : ℝ)=x j rw [←add_div,Int.fract_add_floor] exact mul_div_cancel_left₀ _ hnr.ne' end SingleLatticeCovering.FiniteDimensions namespace SingleLatticeCovering.FiniteDimensions open MeasureTheory Module Set Completion Vertical LatticeGeometry open scoped BigOperators Pointwise theorem finite_dimension_cover {n : ℕ} (hn : 0 < n) (K : Set (Fin n → ℝ)) (hK : IsCompact K) (hconv : Convex ℝ K) (hinterior : (interior K).Nonempty) : ∃ (L : Submodule ℤ (Fin n → ℝ)) (_ : DiscreteTopology L), IsZLattice ℝ L ∧ K+(L : Set (Fin n → ℝ))=univ ∧ (volume K).toReal / ZLattice.covolume L ≤ (2*(n : ℝ))^n := by classical obtain ⟨x₀,hx₀⟩ := hinterior let t := Homeomorph.addRight x₀ let K₀ := t ⁻¹' K have hK₀ : IsCompact K₀ := t.isCompact_preimage.mpr hK have h0 : (0 : Fin n → ℝ) ∈ interior K₀ := by rw [←t.preimage_interior] change 0+x₀ ∈ interior K simpa using hx₀ have hconv₀ : Convex ℝ K₀ := by intro x hx y hy a b ha hb hab have hh := hconv hx hy ha hb hab change a • (x+x₀)+b • (y+x₀) ∈ K at hh change a • x+b • y+x₀ ∈ K have he : a • (x+x₀)+b • (y+x₀) = a • x+b • y+x₀ := by calc _ = (a • x+b • y)+(a+b) • x₀ := by module _ = _ := by rw [hab,one_smul] rwa [he] at hh obtain ⟨b,hb,hbound⟩ := maximal_basis K₀ hK₀ h0 let e := b.equivFun.symm let K' := e ⁻¹' K₀ have hsmall : Icc (0 : Fin n → ℝ) (fun _ => 1/(n : ℝ)) ⊆ K' := convex_basis_cube hn K₀ hconv₀ (interior_subset h0) b hb have hlarge : K' ⊆ Icc (fun _ : Fin n => (-1 : ℝ)) (fun _ => 1) := by intro x hx have hb' := hbound (e x) hx have hj (j : Fin n) : |x j| ≤ 1 := by have hh : |b.equivFun (e x) j| ≤ 1 := hb' j simpa only [e,LinearEquiv.apply_symm_apply] using hh exact ⟨fun j => (abs_le.mp (hj j)).1,fun j => (abs_le.mp (hj j)).2⟩ have hnr : (0 : ℝ) < n := by exact_mod_cast hn let L₀ := contractLattice (integerLattice n) (n : ℝ) hnr.ne' have hcover : K'+(L₀ : Set (Fin n → ℝ))=univ := by apply Set.eq_univ_of_univ_subset rw [←integer_cube_cover hn] exact Set.add_subset_add_right hsmall have hvolume : (volume K').toReal ≤ (2 : ℝ)^n := by have hh := ENNReal.toReal_mono (isCompact_Icc.measure_ne_top (μ := volume)) (measure_mono hlarge) rw [Real.volume_Icc_pi_toReal (by intro j; norm_num)] at hh norm_num at hh exact hh have hcov : ZLattice.covolume L₀ = (n : ℝ)⁻¹^n := by rw [contractLattice_covolume _ _ hnr,integerLattice_covolume,mul_one] have hdens : (volume K').toReal / ZLattice.covolume L₀ ≤ (2*(n : ℝ))^n := by rw [hcov,inv_pow,div_inv_eq_mul,mul_pow] exact mul_le_mul_of_nonneg_right hvolume (pow_nonneg hnr.le _) have himage : (fun x => e x+x₀) '' K'=K := by ext x constructor · rintro ⟨y,hy,rfl⟩ exact hy · intro hx refine ⟨e.symm (x-x₀),?_,?_⟩ · change e (e.symm (x-x₀))+x₀ ∈ K simpa using hx · simp refine ⟨image L₀ e,inferInstance,inferInstance,?_,?_⟩ · rw [←himage] exact affine_image_cover L₀ e x₀ K' hcover · rw [←himage,affine_image_density] exact hdens end SingleLatticeCovering.FiniteDimensions end end end end OAI