import OAI.Geometry.LatticeCovering.Covering namespace OAI section section noncomputable section open Set Metric MeasureTheory MeasureTheory.Measure Module open scoped ENNReal NNReal BigOperators noncomputable section open Set Metric MeasureTheory MeasureTheory.Measure Module open scoped Pointwise ENNReal NNReal BigOperators noncomputable section namespace SingleLatticeCovering.Sections open Set Module MeasureTheory open scoped BigOperators Pointwise variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] lemma affine_section_of_ball_projection {D : ℕ} (hD : 0 < D) (b : Basis (Fin D) ℝ F) (hunit : ∀ i, ‖b i‖ ≤ 1) (hcoord : ∀ y : F, ∀ i, |b.repr y i| ≤ ‖y‖) {K : Set (E × F)} (hconv : Convex ℝ K) {R : ℝ} (hR : 0 < R) (hproj : ∀ v : F, ‖v‖ ≤ 6*(D : ℝ)*R → ∃ x : E, (x,v) ∈ K) : ∃ T : F →ᵃ[ℝ] E, ∀ y : F, ‖y‖ ≤ 2*R → (T y,y) ∈ K := by classical have hDr : (0 : ℝ) < D := by exact_mod_cast hD let c : ℝ := 4*R*D have hc : 0 < c := by dsimp [c]; positivity let v₀ : F := ∑ i : Fin D, (-2*R) • b i let v : Fin D → F := fun i => v₀+c • b i have hv₀ : ‖v₀‖ ≤ 2*R*D := by calc _ ≤ ∑ i : Fin D, ‖(-2*R) • b i‖ := norm_sum_le _ _ _ ≤ ∑ i : Fin D, 2*R := by apply Finset.sum_le_sum; intro i hi rw [norm_smul,Real.norm_eq_abs,abs_mul,abs_of_nonneg hR.le] norm_num nlinarith [hunit i] _ = _ := by simp; ring have hvi : ∀ i, ‖v i‖ ≤ 6*(D : ℝ)*R := by intro i calc _ ≤ ‖v₀‖+‖c • b i‖ := norm_add_le _ _ _ ≤ 2*R*D+c := add_le_add hv₀ (by rw [norm_smul,Real.norm_of_nonneg hc.le]; nlinarith [hunit i]) _ = _ := by dsimp [c]; ring obtain ⟨x₀,hx₀⟩ := hproj v₀ (by nlinarith) choose x hx using fun i => hproj (v i) (hvi i) let lam : F → Fin D → ℝ := fun y i => (b.repr y i+2*R)/c let L : F →ₗ[ℝ] E := ∑ i : Fin D, (b.coord i).smulRight (c⁻¹ • (x i-x₀)) let t₀ : E := x₀+∑ i : Fin D, (2*R/c) • (x i-x₀) let T : F →ᵃ[ℝ] E := L.toAffineMap+AffineMap.const ℝ F t₀ have hT : ∀ y, T y=x₀+∑ i : Fin D, lam y i • (x i-x₀) := by intro y dsimp [T,L,t₀,lam] simp only [ LinearMap.sum_apply,LinearMap.smulRight_apply,Basis.coord_apply,smul_smul] rw [add_comm _ (x₀+_),add_assoc]; congr 1 rw [←Finset.sum_add_distrib] apply Finset.sum_congr rfl; intro i hi rw [←add_smul] congr 1; field_simp; ring refine ⟨T,fun y hy => ?_⟩ have hlam0 : ∀ i, 0 ≤ lam y i := by intro i have hh := (abs_le.mp ((hcoord y i).trans hy)).1 exact div_nonneg (by linarith) hc.le have hlamsum : ∑ i : Fin D, lam y i ≤ 1 := by calc _ ≤ ∑ i : Fin D, 4*R/c := by apply Finset.sum_le_sum; intro i hi apply div_le_div_of_nonneg_right _ hc.le have hh := (abs_le.mp ((hcoord y i).trans hy)).2 linarith _ = 1 := by simp [c]; field_simp let w : Option (Fin D) → ℝ := fun o => o.elim (1-∑ i,lam y i) (lam y) let z : Option (Fin D) → E × F := fun o => o.elim (x₀,v₀) (fun i => (x i,v i)) have hw0 : ∀ o ∈ (Finset.univ : Finset (Option (Fin D))), 0 ≤ w o := by intro o ho; cases o with | none => exact sub_nonneg.mpr hlamsum | some i => exact hlam0 i have hw1 : ∑ o : Option (Fin D), w o=1 := by simp [w,Fintype.sum_option] have hz : ∀ o ∈ (Finset.univ : Finset (Option (Fin D))), z o ∈ K := by intro o ho; cases o with | none => exact hx₀ | some i => exact hx i have hmem := hconv.sum_mem hw0 hw1 hz have he : ∑ o : Option (Fin D), w o • z o=(T y,y) := by rw [Fintype.sum_option] dsimp only [w,z,Option.elim] apply Prod.ext · simp only [Prod.fst_add,Prod.fst_sum,Prod.smul_fst] change (1-∑ i,lam y i) • x₀+∑ i,lam y i • x i=T y rw [hT] simp only [smul_sub,Finset.sum_sub_distrib,←Finset.sum_smul] module · simp only [Prod.snd_add,Prod.snd_sum,Prod.smul_snd] change (1-∑ i,lam y i) • v₀+∑ i,lam y i • v i=y dsimp only [v] simp only [smul_add,smul_smul,Finset.sum_add_distrib,←Finset.sum_smul] have hh : ∀ i, lam y i*c=b.repr y i+2*R := by intro i; dsimp [lam]; field_simp simp_rw [hh,add_smul] rw [Finset.sum_add_distrib,b.sum_repr] have hsum : v₀+(∑ i : Fin D, (2*R) • b i)=0 := by dsimp only [v₀] rw [←Finset.sum_add_distrib] simp only [←add_smul,neg_mul,neg_add_cancel,zero_smul,Finset.sum_const_zero] calc _ = y+(v₀+∑ i : Fin D, (2*R) • b i) := by module _ = y := by rw [hsum,add_zero] rwa [he] at hmem end SingleLatticeCovering.Sections noncomputable section namespace SingleLatticeCovering.Sections open Set Metric MeasureTheory MeasureTheory.Measure Module open scoped Pointwise ENNReal NNReal BigOperators variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ E] [FiniteDimensional ℝ F] [MeasurableSpace E] [BorelSpace E] [MeasurableSpace F] [BorelSpace F] def gaussian (F : Type*) [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F] (y : F) : ℝ := (Real.sqrt (2*Real.pi))^(-(finrank ℝ F : ℤ))*Real.exp (-‖y‖^2/2) lemma gaussian_pos {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F] [MeasurableSpace F] [BorelSpace F] (y : F) : 0 < gaussian F y := by dsimp [gaussian] exact mul_pos (zpow_pos (Real.sqrt_pos.mpr (by positivity)) _) (Real.exp_pos _) lemma gaussian_near {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ F] [MeasurableSpace F] [BorelSpace F] {R : ℝ} (hR : 0 ≤ R) {m : ℕ} (hm : 0 < m) {a y : F} (ha : ‖a‖ ≤ R) (hy : ‖y‖ ≤ R) (hya : dist y a ≤ R/(m : ℝ)^2) (hsmall : R^2/(m : ℝ)^2 ≤ 1) : Real.exp (-1)*gaussian F y ≤ gaussian F a ∧ gaussian F a ≤ Real.exp 1*gaussian F y := by have hm0 : (0 : ℝ) < m := by exact_mod_cast hm have hdist : |‖a‖-‖y‖| ≤ R/(m : ℝ)^2 := (abs_norm_sub_norm_le a y).trans (by simpa only [dist_comm,dist_eq_norm] using hya) have hsq : |‖a‖^2-‖y‖^2| ≤ 2 := by have he : ‖a‖^2-‖y‖^2=(‖a‖-‖y‖)*(‖a‖+‖y‖) := by ring rw [he,abs_mul,abs_of_nonneg (add_nonneg (norm_nonneg _) (norm_nonneg _))] calc _ ≤ (R/(m : ℝ)^2)*(2*R) := mul_le_mul hdist (by linarith) (by positivity) (by positivity) _ = 2*(R^2/(m : ℝ)^2) := by ring _ ≤ 2 := by linarith have hp := abs_le.mp hsq have hlow : -1+-‖y‖^2/2 ≤ -‖a‖^2/2 := by linarith have hupp : -‖a‖^2/2 ≤ 1+-‖y‖^2/2 := by linarith have hb : 0 ≤ (Real.sqrt (2*Real.pi))^(-(finrank ℝ F : ℤ)) := by positivity constructor · have hh := mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr hlow) hb simpa only [Real.exp_add,gaussian,mul_left_comm] using hh · have hh := mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr hupp) hb simpa only [Real.exp_add,gaussian,mul_left_comm] using hh theorem finite_section_labels (μ : Measure E) [IsAddHaarMeasure μ] (ν : Measure F) [IsAddHaarMeasure ν] {K : Set (E × F)} (hK : IsCompact K) (hconv : Convex ℝ K) {R V : ℝ} (hR : 0 < R) (hV : 0 < V) (hm : 2 ≤ finrank ℝ E) (hcenter : ∀ z : F, ‖z‖ ≤ 2*R → (0,z) ∈ K) (hlower : ∀ y : F, ‖y‖ ≤ R → V*gaussian F y/2 ≤ (μ (fiber K y)).toReal) (hsmall : R^2/(finrank ℝ E : ℝ)^2 ≤ 1) : ∃ S : Finset F, (S.card : ℝ) ≤ (3*(finrank ℝ E : ℝ)^2)^(finrank ℝ F) ∧ ∃ J : F → Set E, (∀ a ∈ S, IsCompact (J a) ∧ Convex ℝ (J a) ∧ (0 : E) ∈ J a ∧ (μ (J a)).toReal=V*gaussian F a/8) ∧ ∀ y : F, ‖y‖ ≤ R → ∃ a ∈ S, J a ⊆ fiber K y ∧ (Real.exp (-1)/8)*V*gaussian F y ≤ (μ (J a)).toReal ∧ (μ (J a)).toReal ≤ (Real.exp 1/8)*V*gaussian F y := by classical obtain ⟨S,hSc,hSn,hnet⟩ := ball_finite_net ν hR (show 1 ≤ finrank ℝ E by omega) let d : ℝ := 1-(finrank ℝ E : ℝ)^(-2 : ℤ) have hn2 : (2 : ℝ) ≤ finrank ℝ E := by exact_mod_cast hm have hd : 0 ≤ d := by dsimp [d]; norm_num only [zpow_neg, zpow_ofNat] have hh : 1 ≤ (finrank ℝ E : ℝ)^2 := by nlinarith exact sub_nonneg.mpr ((inv_le_one₀ (by positivity)).mpr hh) have hdata : ∀ a ∈ S, ∃ Ja : Set E, IsCompact Ja ∧ Convex ℝ Ja ∧ (0 : E) ∈ Ja ∧ Ja ⊆ d • fiber K a ∧ (μ Ja).toReal=V*gaussian F a/8 := by intro a ha have haR := hSn a ha have hzero : (0 : E) ∈ fiber K a := hcenter a (by linarith) have hzero' : (0 : E) ∈ d • fiber K a := ⟨0,hzero,smul_zero _⟩ apply prescribed_volume μ (by omega) ((fiber_compact hK a).smul d) ((fiber_convex hconv a).smul d) hzero' (by positivity [gaussian_pos a]) rw [μ.addHaar_smul_of_nonneg hd,ENNReal.toReal_mul,ENNReal.toReal_ofReal (pow_nonneg hd _)] have hh := shrink_power_half hm have hv := hlower a haR have hgp := gaussian_pos a have hhalf : (1/2 : ℝ)*(V*gaussian F a/2) ≤ d^finrank ℝ E*(μ (fiber K a)).toReal := mul_le_mul hh hv (by positivity) (pow_nonneg hd _) nlinarith have hex : ∀ a : F, ∃ Ja : Set E, a ∈ S → IsCompact Ja ∧ Convex ℝ Ja ∧ (0 : E) ∈ Ja ∧ Ja ⊆ d • fiber K a ∧ (μ Ja).toReal=V*gaussian F a/8 := by intro a by_cases ha : a ∈ S · exact ⟨(hdata a ha).choose,fun _ => (hdata a ha).choose_spec⟩ · exact ⟨∅,fun hh => (ha hh).elim⟩ choose J hJ using hex refine ⟨S,hSc,J,?_,?_⟩ · intro a ha; exact ⟨(hJ a ha).1,(hJ a ha).2.1,(hJ a ha).2.2.1,(hJ a ha).2.2.2.2⟩ · intro y hy obtain ⟨a,ha,hya⟩ := hnet y hy have hsub := nearby_fiber_inclusion hconv hR.le (show 1 ≤ finrank ℝ E by omega) hcenter (hSn a ha) hya have hg := gaussian_near hR.le (show 0 < finrank ℝ E by omega) (hSn a ha) hy hya hsmall refine ⟨a,ha,(hJ a ha).2.2.2.1.trans hsub,?_,?_⟩ <;> rw [(hJ a ha).2.2.2.2] · nlinarith [mul_le_mul_of_nonneg_left hg.1 hV.le] · nlinarith [mul_le_mul_of_nonneg_left hg.2 hV.le] end SingleLatticeCovering.Sections noncomputable section namespace SingleLatticeCovering.Sections open Set MeasureTheory MeasureTheory.Measure Module open scoped Pointwise ENNReal BigOperators variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] def sectionShear (T : F →ᵃ[ℝ] E) : (E × F) ≃ᵃ[ℝ] (E × F) := AffineEquiv.ofBijective (φ := ((AffineMap.fst : (E × F) →ᵃ[ℝ] E)-T.comp AffineMap.snd).prod AffineMap.snd) (by constructor · rintro ⟨x,y⟩ ⟨x',y'⟩ h have hy : y=y' := congrArg Prod.snd h subst y' have hx : x-T y=x'-T y := congrArg Prod.fst h exact Prod.ext (by simpa using congrArg (fun z => z+T y) hx) rfl · rintro ⟨x,y⟩ refine ⟨(x+T y,y),?_⟩ change (x+T y-T y,y)=(x,y) simp) @[simp] lemma sectionShear_apply (T : F →ᵃ[ℝ] E) (x : E) (y : F) : sectionShear T (x,y)=(x-T y,y) := rfl lemma fiber_sectionShear (T : F →ᵃ[ℝ] E) (K : Set (E × F)) (y : F) : fiber (sectionShear T '' K) y=(fun x => x-T y) '' fiber K y := by ext x; constructor · rintro ⟨⟨u,v⟩,hK,he⟩ have hv : v=y := congrArg Prod.snd he subst v exact ⟨u,hK,congrArg Prod.fst he⟩ · rintro ⟨u,hu,rfl⟩ exact ⟨(u,y),hu,rfl⟩ variable [FiniteDimensional ℝ E] [FiniteDimensional ℝ F] [MeasurableSpace E] [BorelSpace E] [MeasurableSpace F] [BorelSpace F] lemma fiber_sectionShear_measure {E : Type*} {F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ E] [FiniteDimensional ℝ F] [MeasurableSpace E] [BorelSpace E] [MeasurableSpace F] [BorelSpace F] (μ : Measure E) [IsAddHaarMeasure μ] (T : F →ᵃ[ℝ] E) (K : Set (E × F)) (y : F) : μ (fiber (sectionShear T '' K) y)=μ (fiber K y) := by rw [fiber_sectionShear] simp only [sub_eq_add_neg,Set.image_add_right,measure_preimage_add_right] lemma sectionShear_measure (μ : Measure E) [IsAddHaarMeasure μ] (ν : Measure F) [IsAddHaarMeasure ν] (T : F →ᵃ[ℝ] E) {K : Set (E × F)} (hK : IsCompact K) : (μ.prod ν) (sectionShear T '' K)=(μ.prod ν) K := by have hc := (sectionShear T).toAffineMap.continuous_of_finiteDimensional have hK' : IsCompact (sectionShear T '' K) := hK.image hc rw [Measure.prod_apply_symm hK'.measurableSet,Measure.prod_apply_symm hK.measurableSet] apply lintegral_congr intro y exact fiber_sectionShear_measure μ T K y lemma convex_interior_of_measure_pos (μ : Measure E) [IsAddHaarMeasure μ] {J : Set E} (hconv : Convex ℝ J) (hpos : 0 < (μ J).toReal) : (interior J).Nonempty := by apply hconv.interior_nonempty_iff_affineSpan_eq_top.mpr by_contra h have hn : μ J=0 := measure_mono_null (subset_affineSpan ℝ J) (μ.addHaar_affineSubspace _ h) rw [hn,ENNReal.toReal_zero] at hpos exact (lt_irrefl 0) hpos theorem centered_affine_image {D : ℕ} (hD : 0 < D) (b : Basis (Fin D) ℝ F) (hunit : ∀ i, ‖b i‖ ≤ 1) (hcoord : ∀ y : F, ∀ i, |b.repr y i| ≤ ‖y‖) (μ : Measure E) [IsAddHaarMeasure μ] (ν : Measure F) [IsAddHaarMeasure ν] {K : Set (E × F)} (hK : IsCompact K) (hconv : Convex ℝ K) {R : ℝ} (hR : 0 < R) (hproj : ∀ v : F, ‖v‖ ≤ 6*(D : ℝ)*R → ∃ x : E, (x,v) ∈ K) : ∃ e : (E × F) ≃ᵃ[ℝ] (E × F), IsCompact (e '' K) ∧ Convex ℝ (e '' K) ∧ (μ.prod ν) (e '' K)=(μ.prod ν) K ∧ (∀ y, μ (fiber (e '' K) y)=μ (fiber K y)) ∧ (∀ y : F, ‖y‖ ≤ 2*R → (0,y) ∈ e '' K) := by obtain ⟨T,hT⟩ := affine_section_of_ball_projection hD b hunit hcoord hconv hR hproj refine ⟨sectionShear T,hK.image (sectionShear T).toAffineMap.continuous_of_finiteDimensional, hconv.affine_image (sectionShear T).toAffineMap, sectionShear_measure μ ν T hK,fun y => fiber_sectionShear_measure μ T K y,?_⟩ intro y hy refine ⟨(T y,y),hT y hy,?_⟩ simp end SingleLatticeCovering.Sections noncomputable section namespace SingleLatticeCovering.Sections open Set MeasureTheory MeasureTheory.Measure Filter Topology open scoped ENNReal variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] lemma eventually_fiber_subset {E : Type*} {F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {K : Set (E × F)} (hK : IsCompact K) {y : F} {U : Set E} (hU : IsOpen U) (hyU : fiber K y ⊆ U) : ∀ᶠ z in 𝓝 y, fiber K z ⊆ U := by let A : Set F := Prod.snd '' (K ∩ Prod.fst ⁻¹' Uᶜ) have hA : IsClosed A := ((hK.inter_right (hU.isClosed_compl.preimage continuous_fst)).image continuous_snd).isClosed have hy : y ∈ Aᶜ := by rintro ⟨⟨x,z⟩,⟨hx,hxU⟩,he⟩ change z=y at he subst z exact hxU (hyU hx) filter_upwards [hA.isOpen_compl.mem_nhds hy] with z hz intro x hx by_contra hxU exact hz ⟨(x,z),⟨hx,hxU⟩,rfl⟩ lemma upperSemicontinuous_fiber_measure [MeasurableSpace E] [BorelSpace E] (μ : Measure E) [OuterRegular μ] {K : Set (E × F)} (hK : IsCompact K) : UpperSemicontinuous (fun y : F => μ (fiber K y)) := by intro y r hyr obtain ⟨U,hfU,hU,hμU⟩ := (fiber K y).exists_isOpen_lt_of_lt r hyr filter_upwards [eventually_fiber_subset hK hU hfU] with z hz exact (measure_mono hz).trans_lt hμU lemma lower_bound_everywhere_of_ae {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F] [MeasurableSpace F] [BorelSpace F] (ν : Measure F) [IsOpenPosMeasure ν] {f g : F → ℝ≥0∞} (hf : UpperSemicontinuous f) (hg : Continuous g) {U : Set F} (hU : IsOpen U) (hae : ∀ᵐ y ∂ν, y ∈ U → g y ≤ f y) : ∀ y ∈ U, g y ≤ f y := by intro y hy by_contra h have hfg : f y < g y := lt_of_not_ge h obtain ⟨q,hfq,hqg⟩ := exists_between hfg let V := U ∩ (f ⁻¹' Iio q) ∩ (g ⁻¹' Ioi q) have hV : IsOpen V := (hU.inter (hf.isOpen_preimage q)).inter (isOpen_Ioi.preimage hg) have hyV : y ∈ V := ⟨⟨hy,hfq⟩,hqg⟩ obtain ⟨z,hz,hzV⟩ := (ν.dense_of_ae hae).exists_mem_open hV ⟨y,hyV⟩ exact (not_lt_of_ge (hz hzV.1.1)) (hzV.1.2.trans hzV.2) theorem canonical_marginal_lower [MeasurableSpace E] [BorelSpace E] [MeasurableSpace F] [BorelSpace F] (μ : Measure E) [OuterRegular μ] (ν : Measure F) [IsOpenPosMeasure ν] {K : Set (E × F)} (hK : IsCompact K) {g : F → ℝ≥0∞} (hg : Continuous g) {S : ℝ} (hae : ∀ᵐ y ∂ν, ‖y‖ < S → g y ≤ μ (fiber K y)) : ∀ y : F, ‖y‖ < S → g y ≤ μ (fiber K y) := by exact lower_bound_everywhere_of_ae ν (upperSemicontinuous_fiber_measure μ hK) hg (isOpen_lt continuous_norm continuous_const) hae end SingleLatticeCovering.Sections noncomputable section namespace SingleLatticeCovering.Sections open Set Metric MeasureTheory MeasureTheory.Measure Module open scoped Pointwise ENNReal BigOperators variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] [FiniteDimensional ℝ E] [FiniteDimensional ℝ F] [MeasurableSpace E] [BorelSpace E] [MeasurableSpace F] [BorelSpace F] theorem gaussian_marginal_to_sections {D : ℕ} (hD : 0 < D) (b : Basis (Fin D) ℝ F) (hunit : ∀ i, ‖b i‖ ≤ 1) (hcoord : ∀ y : F, ∀ i, |b.repr y i| ≤ ‖y‖) (μ : Measure E) [IsAddHaarMeasure μ] (ν : Measure F) [IsAddHaarMeasure ν] {K : Set (E × F)} (hK : IsCompact K) (hconv : Convex ℝ K) {R : ℝ} (hR : 0 < R) (hV : 0 < ((μ.prod ν) K).toReal) (hm : 2 ≤ finrank ℝ E) (hlower : ∀ y : F, ‖y‖ ≤ 6*(D : ℝ)*R → ((μ.prod ν) K).toReal*gaussian F y/2 ≤ (μ (fiber K y)).toReal) (hsmall : R^2/(finrank ℝ E : ℝ)^2 ≤ 1) : ∃ e : (E × F) ≃ᵃ[ℝ] (E × F), IsCompact (e '' K) ∧ Convex ℝ (e '' K) ∧ (μ.prod ν) (e '' K)=(μ.prod ν) K ∧ ∃ S : Finset F, (S.card : ℝ) ≤ (3*(finrank ℝ E : ℝ)^2)^D ∧ ∃ J : F → Set E, (∀ a ∈ S, IsCompact (J a) ∧ Convex ℝ (J a) ∧ (interior (J a)).Nonempty) ∧ ∀ y : F, ‖y‖ ≤ R → ∃ a ∈ S, J a ⊆ fiber (e '' K) y ∧ (Real.exp (-1)/8)*((μ.prod ν) (e '' K)).toReal*gaussian F y ≤ (μ (J a)).toReal ∧ (μ (J a)).toReal ≤ (Real.exp 1/8)*((μ.prod ν) (e '' K)).toReal*gaussian F y := by have hDr : (1 : ℝ) ≤ D := by exact_mod_cast (show 1 ≤ D by omega) obtain ⟨e,heK,heconv,heμ,hef,hezero⟩ := centered_affine_image hD b hunit hcoord μ ν hK hconv hR (by intro y hy have hp : 0 < (μ (fiber K y)).toReal := (by positivity [gaussian_pos y] : 0 < ((μ.prod ν) K).toReal*gaussian F y/2).trans_le (hlower y hy) by_contra hn push Not at hn have hempty : fiber K y=∅ := Set.eq_empty_iff_forall_notMem.mpr hn rw [hempty,measure_empty,ENNReal.toReal_zero] at hp exact (lt_irrefl 0) hp) obtain ⟨S,hSc,J,hJ,hlabels⟩ := finite_section_labels μ ν heK heconv hR hV hm hezero (by intro y hy rw [hef] exact hlower y (hy.trans (by nlinarith))) hsmall have hdim : finrank ℝ F=D := by simpa using Module.finrank_eq_card_basis b refine ⟨e,heK,heconv,heμ,S,by simpa only [hdim] using hSc,J,?_,?_⟩ · intro a ha refine ⟨(hJ a ha).1,(hJ a ha).2.1,convex_interior_of_measure_pos μ (hJ a ha).2.1 ?_⟩ rw [(hJ a ha).2.2.2] positivity [gaussian_pos a] · intro y hy simpa only [heμ] using hlabels y hy end SingleLatticeCovering.Sections end end end end end end end end section noncomputable section namespace SingleLatticeCovering.Sections open MeasureTheory open scoped BigOperators lemma gaussian_toLp {D : ℕ} (y : Fin D → ℝ) : gaussian (EuclideanSpace ℝ (Fin D)) (WithLp.toLp 2 y)=Folded.gamma y := by classical unfold gaussian Folded.gamma Folded.gamma1 rw [finrank_euclideanSpace_fin,EuclideanSpace.real_norm_sq_eq] rw [Finset.prod_div_distrib,Finset.prod_const,Finset.card_univ,Fintype.card_fin] rw [← Real.exp_sum,zpow_neg,zpow_natCast] rw [div_eq_mul_inv,mul_comm] congr 2 simp only [div_eq_mul_inv,Finset.sum_mul,Finset.sum_neg_distrib,neg_mul] end SingleLatticeCovering.Sections end end section noncomputable section namespace SingleLatticeCovering.Sections open MeasureTheory LatticeGeometry open scoped BigOperators lemma appendEquiv_measurePreserving (m D : ℕ) : MeasurePreserving (appendEquiv m D) ((volume : Measure (Fin m → ℝ)).prod (volume : Measure (Fin D → ℝ))) volume := by have h := (volume_preserving_arrowCongr' (finSumFinEquiv (m := m) (n := D)) (MeasurableEquiv.refl ℝ) (MeasurePreserving.id volume)).comp (volume_measurePreserving_sumPiEquivProdPi_symm (fun _ : Fin m ⊕ Fin D => ℝ)) have he : (fun t : (Fin m → ℝ) × (Fin D → ℝ) => (MeasurableEquiv.arrowCongr' finSumFinEquiv (MeasurableEquiv.refl ℝ)) ((MeasurableEquiv.sumPiEquivProdPi (fun _ : Fin m ⊕ Fin D => ℝ)).symm t))= appendEquiv m D := by funext t i refine Fin.addCases ?_ ?_ i <;> intro j <;> simp [MeasurableEquiv.arrowCongr',MeasurableEquiv.sumPiEquivProdPi, Equiv.arrowCongr',appendEquiv,Fin.appendEquiv] exact he ▸ h def flatten (m D : ℕ) : ((Fin m → ℝ) × EuclideanSpace ℝ (Fin D)) ≃ₗ[ℝ] (Fin (m+D) → ℝ) := ((LinearEquiv.refl ℝ (Fin m → ℝ)).prodCongr (WithLp.linearEquiv 2 ℝ (Fin D → ℝ))).trans (appendEquiv m D) @[simp] lemma flatten_apply (m D : ℕ) (x : Fin m → ℝ) (y : EuclideanSpace ℝ (Fin D)) : flatten m D (x,y)=Fin.append x (WithLp.ofLp y) := rfl lemma flatten_measurePreserving (m D : ℕ) : MeasurePreserving (flatten m D) volume volume := (appendEquiv_measurePreserving m D).comp ((MeasurePreserving.id (volume : Measure (Fin m → ℝ))).prod (PiLp.volume_preserving_ofLp (Fin D))) lemma flatten_volume (m D : ℕ) {K : Set ((Fin m → ℝ) × EuclideanSpace ℝ (Fin D))} (hK : IsCompact K) : volume (flatten m D '' K)=volume K := by have h := (flatten_measurePreserving m D).measure_preimage ((hK.image (flatten m D).toContinuousLinearEquiv.continuous).measurableSet.nullMeasurableSet) change volume ((flatten m D) ⁻¹' ((flatten m D) '' K))=volume ((flatten m D) '' K) at h rw [(flatten m D).injective.preimage_image] at h exact h.symm end SingleLatticeCovering.Sections end end section noncomputable section namespace SingleLatticeCovering.Sections open MeasureTheory Set open scoped BigOperators def coordinateFiber {m D : ℕ} (K : Set (Fin (m+D) → ℝ)) (y : Fin D → ℝ) : Set (Fin m → ℝ) := {x | Fin.append x y ∈ K} lemma flatten_fiber {m D : ℕ} (K : Set (Fin (m+D) → ℝ)) (y : EuclideanSpace ℝ (Fin D)) : fiber ((flatten m D).symm '' K) y=coordinateFiber K (WithLp.ofLp y) := by ext x change (x,y) ∈ (flatten m D).symm '' K ↔ _ constructor · rintro ⟨z,hz,hzxy⟩ change flatten m D (x,y) ∈ K rw [←hzxy,(flatten m D).apply_symm_apply] exact hz · intro hx exact ⟨flatten m D (x,y),hx,(flatten m D).symm_apply_apply _⟩ theorem coordinate_marginal_to_labels {m D : ℕ} (hm : 2 ≤ m) (hD : 0 < D) {K : Set (Fin (m+D) → ℝ)} (hK : IsCompact K) (hconv : Convex ℝ K) (hV : 0 < (volume K).toReal) {R : ℝ} (hR : 0 < R) (hsmall : R^2/(m : ℝ)^2 ≤ 1) (hlower : ∀ y : Fin D → ℝ, ‖WithLp.toLp 2 y‖ ≤ 6*(D : ℝ)*R → (volume K).toReal*Folded.gamma y/2 ≤ (volume (coordinateFiber K y)).toReal) : ∃ e : (Fin (m+D) → ℝ) ≃ᵃ[ℝ] (Fin (m+D) → ℝ), IsCompact (e '' K) ∧ Convex ℝ (e '' K) ∧ volume (e '' K)=volume K ∧ ∃ S : Finset (EuclideanSpace ℝ (Fin D)), (S.card : ℝ) ≤ (3*(m : ℝ)^2)^D ∧ ∃ J : EuclideanSpace ℝ (Fin D) → Set (Fin m → ℝ), (∀ a ∈ S, IsCompact (J a) ∧ Convex ℝ (J a) ∧ (interior (J a)).Nonempty) ∧ ∀ y : Fin D → ℝ, (∑ j, (y j)^2) ≤ R^2 → ∃ a ∈ S, (∀ x ∈ J a, Fin.append x y ∈ e '' K) ∧ (Real.exp (-1)/8)*(volume (e '' K)).toReal*Folded.gamma y ≤ (volume (J a)).toReal ∧ (volume (J a)).toReal ≤ (Real.exp 1/8)*(volume (e '' K)).toReal*Folded.gamma y := by let f := flatten m D let Kp : Set ((Fin m → ℝ) × EuclideanSpace ℝ (Fin D)) := f.symm '' K have hKp : IsCompact Kp := hK.image f.symm.toContinuousLinearEquiv.continuous have hconvp : Convex ℝ Kp := hconv.linear_image f.symm.toLinearMap have hflat : f '' Kp=K := by dsimp [Kp]; exact f.image_symm_image K have hvolp : volume Kp=volume K := by rw [←flatten_volume m D hKp]; exact congrArg volume hflat let b := EuclideanSpace.basisFun (Fin D) ℝ have hunit : ∀ i, ‖b.toBasis i‖ ≤ 1 := by intro i; exact (b.orthonormal.1 i).le have hcoord : ∀ y : EuclideanSpace ℝ (Fin D), ∀ i, |b.toBasis.repr y i| ≤ ‖y‖ := by intro y i rw [OrthonormalBasis.coe_toBasis_repr_apply] simpa only [Real.norm_eq_abs,b,EuclideanSpace.basisFun_repr] using PiLp.norm_apply_le y i obtain ⟨e,heK,heconv,hevol,S,hSc,J,hJ,hlabels⟩ := gaussian_marginal_to_sections hD b.toBasis hunit hcoord (volume : Measure (Fin m → ℝ)) (volume : Measure (EuclideanSpace ℝ (Fin D))) hKp hconvp hR (by simpa only [←Measure.volume_eq_prod,hvolp] using hV) (by simpa using hm) (by intro y hy have hh := hlower y.ofLp (by simpa using hy) simpa only [←Measure.volume_eq_prod,hvolp,Kp,f,flatten_fiber,←gaussian_toLp,WithLp.toLp_ofLp] using hh) (by simpa using hsmall) let e' : (Fin (m+D) → ℝ) ≃ᵃ[ℝ] (Fin (m+D) → ℝ) := (f.symm.toAffineEquiv.trans e).trans f.toAffineEquiv have he'image : e' '' K=f '' (e '' Kp) := by change (fun x => f (e (f.symm x))) '' K = _ simp only [Kp,Set.image_image] have hevol' : volume (e '' Kp)=volume Kp := by simpa only [←Measure.volume_eq_prod] using hevol have he'vol : volume (e' '' K)=volume K := by rw [he'image,flatten_volume m D heK] exact hevol'.trans hvolp refine ⟨e',?_,?_,he'vol,S,?_,J,hJ,?_⟩ · rw [he'image]; exact heK.image f.toContinuousLinearEquiv.continuous · rw [he'image]; exact heconv.linear_image f.toLinearMap · simpa using hSc · intro y hy have hynorm : ‖WithLp.toLp 2 y‖ ≤ R := by apply (sq_le_sq₀ (norm_nonneg _) hR.le).mp simpa only [EuclideanSpace.real_norm_sq_eq] using hy obtain ⟨a,ha,hsub,hl,hu⟩ := hlabels (WithLp.toLp 2 y) hynorm refine ⟨a,ha,?_,?_,?_⟩ · intro x hx rw [he'image] exact ⟨(x,WithLp.toLp 2 y),hsub hx,rfl⟩ · rw [←gaussian_toLp,he'vol,←hvolp] simpa only [←Measure.volume_eq_prod,hevol'] using hl · rw [←gaussian_toLp,he'vol,←hvolp] simpa only [←Measure.volume_eq_prod,hevol'] using hu end SingleLatticeCovering.Sections end end section noncomputable section open Real Filter Topology Asymptotics end end end end OAI