| import Mathlib |
| import OAI.GroupTheory.SimpleAmenable.RandomFields.FlagAveragingMatrices |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.FullGroupAffineData |
|
|
| namespace OAI |
|
|
| section |
| section |
| open scoped symmDiff |
| namespace SimpleAmenable |
| open scoped commutatorElement |
| open scoped commutatorElement |
| section FlagSignalEnergy |
| open Classical |
|
|
| noncomputable def flagMeanSignal {a m D : ℕ} {v : ℝ×ℝ} |
| (θ : (ℝ×ℝ) → ℝ) (N : ℝ) (z : FlagSite a m D v) : ℝ := |
| θ (scaledSiteConjugate N z) |
|
|
| noncomputable def flagSignalDifference {a m D : ℕ} {v : ℝ×ℝ} (hD : 0<D) |
| (g : polygonFullGroup a m) (θ : (ℝ×ℝ) → ℝ) (N : ℝ) (z : FlagSite a m D v) : ℝ := |
| flagMeanSignal θ N (flagSiteAction hD g⁻¹ z)-flagMeanSignal θ N z |
|
|
| theorem flagSignalDifference_chart {a m D : ℕ} {v : ℝ×ℝ} (hD : 0<D) |
| (g : polygonFullGroup a m) (θ : (ℝ×ℝ) → ℝ) (N : ℝ) (z : FlagSite a m D v) : |
| flagSignalDifference hD g θ N z= |
| θ ((flagSiteConjugate z).1/N+conjugate (flagAffineData g⁻¹ z.val.1 z.val.2).2.1/N, |
| (flagSiteConjugate z).2/N+conjugate (flagAffineData g⁻¹ z.val.1 z.val.2).2.2/N)- |
| θ (scaledSiteConjugate N z) := by |
| simp only [flagSignalDifference,flagMeanSignal,scaledSiteConjugate, |
| flagAffineData_conjugate hD,Prod.fst_add,Prod.snd_add,add_div] |
|
|
| theorem flagSiteAction_displacement {a m D : ℕ} {v : ℝ×ℝ} (hD : 0<D) |
| (g : polygonFullGroup a m) : |
| ∃Q : ℝ,0≤Q ∧ ∀z : FlagSite a m D v, |
| |(flagSiteConjugate (flagSiteAction hD g z)).1-(flagSiteConjugate z).1|+ |
| |(flagSiteConjugate (flagSiteAction hD g z)).2-(flagSiteConjugate z).2| ≤ Q := by |
| obtain ⟨Q,hQ,hb⟩ := flagAffineData_bounded (v:=v) g |
| (fun u => |conjugate u.1|+|conjugate u.2|) |
| refine ⟨Q,hQ,fun z => ?_⟩ |
| have hh := hb z.val |
| rw [abs_of_nonneg (add_nonneg (abs_nonneg _) (abs_nonneg _))] at hh |
| simpa only [flagAffineData_conjugate hD,Prod.fst_add,Prod.snd_add,add_sub_cancel_left] using hh |
|
|
| theorem flagSignalDifference_amplitude {a m D : ℕ} {v : ℝ×ℝ} (hD : 0<D) |
| (g : polygonFullGroup a m) (θ : (ℝ×ℝ) → ℝ) {L : ℝ} (hL : 0≤L) |
| (hLip : ∀x y,|θ x-θ y| ≤ L*(|x.1-y.1|+|x.2-y.2|)) : |
| ∃ C : ℝ,0≤C ∧ ∀N : ℝ,0<N → ∀z : FlagSite a m D v, |
| |flagSignalDifference hD g θ N z| ≤ C/N := by |
| obtain ⟨Q,hQ,hdisp⟩ := flagSiteAction_displacement (v:=v) hD g⁻¹ |
| refine ⟨L*Q,mul_nonneg hL hQ,fun N hN z => ?_⟩ |
| have h := hLip (scaledSiteConjugate N (flagSiteAction hD g⁻¹ z)) (scaledSiteConjugate N z) |
| simp only [scaledSiteConjugate,← sub_div,abs_div,abs_of_pos hN,← add_div] at h |
| change |flagSignalDifference hD g θ N z| ≤ _ at h |
| calc |
| _ ≤ _ := h |
| _ ≤ L*(Q/N) := mul_le_mul_of_nonneg_left |
| (div_le_div_of_nonneg_right (hdisp z) hN.le) hL |
| _ = _ := (mul_div_assoc _ _ _).symm |
|
|
| theorem flagMeanSignal_outside {a m D : ℕ} {v : ℝ×ℝ} |
| (θ : (ℝ×ℝ) → ℝ) {q K N : ℝ} (hN : 0<N) |
| (hcompact : ∀x,K ≤ ‖x‖ → θ x=q) (z : FlagSite a m D v) |
| (hz : z∉flagSiteBox (N*K)) : flagMeanSignal θ N z=q := by |
| apply hcompact |
| by_contra hh |
| have hh' : ‖scaledSiteConjugate N z‖<K := lt_of_not_ge hh |
| have h₁ := norm_fst_le (scaledSiteConjugate N z) |
| have h₂ := norm_snd_le (scaledSiteConjugate N z) |
| simp only [scaledSiteConjugate,Real.norm_eq_abs,abs_div,abs_of_pos hN] at h₁ h₂ |
| apply hz |
| constructor |
| · have h := (div_lt_iff₀ hN).mp (h₁.trans_lt hh'); nlinarith |
| · have h := (div_lt_iff₀ hN).mp (h₂.trans_lt hh'); nlinarith |
|
|
| noncomputable def flagSignalSupport {a m D : ℕ} {v : ℝ×ℝ} (hD : 0<D) |
| (g : polygonFullGroup a m) {K N : ℝ} (hK : 0<K) (hN : 0<N) : Finset (FlagSite a m D v) := |
| let S := (flagSiteBox_finite (v:=v) hD (mul_pos hN hK)).toFinset |
| S ∪ S.image (flagSiteAction hD g) |
|
|
| theorem flagSignalDifference_support {a m D : ℕ} {v : ℝ×ℝ} (hD : 0<D) |
| (g : polygonFullGroup a m) (θ : (ℝ×ℝ) → ℝ) {q K N : ℝ} (hK : 0<K) (hN : 0<N) |
| (hcompact : ∀x,K ≤ ‖x‖ → θ x=q) (z : FlagSite a m D v) |
| (hz : z∉flagSignalSupport hD g hK hN) : flagSignalDifference hD g θ N z=0 := by |
| have hpair : z∉flagSiteBox (N*K) ∧ flagSiteAction hD g⁻¹ z∉flagSiteBox (N*K) := by |
| have hn : z∉(flagSiteBox_finite (v:=v) hD (mul_pos hN hK)).toFinset ∧ |
| z∉((flagSiteBox_finite (v:=v) hD (mul_pos hN hK)).toFinset.image (flagSiteAction hD g)) := by |
| simpa only [flagSignalSupport,Finset.mem_union,not_or] using hz |
| refine ⟨?_,?_⟩ |
| · simpa only [Set.Finite.mem_toFinset] using hn.1 |
| · intro hh |
| apply hn.2 |
| apply Finset.mem_image.mpr |
| refine ⟨flagSiteAction hD g⁻¹ z,(Set.Finite.mem_toFinset _).mpr hh,?_⟩ |
| rw [← flagSiteAction_mul,mul_inv_cancel,flagSiteAction_one] |
| simp only [flagSignalDifference,flagMeanSignal_outside θ hN hcompact z hpair.1, |
| flagMeanSignal_outside θ hN hcompact _ hpair.2,sub_self] |
|
|
| theorem flagSignalSupport_card {a m D : ℕ} {v : ℝ×ℝ} (hD : 0<D) |
| (g : polygonFullGroup a m) {K N : ℝ} (hK : 0<K) (hN : 0<N) (hN₁ : 1≤N) : |
| ((flagSignalSupport (v:=v) hD g hK hN).card:ℝ) ≤ |
| 2*(m:ℝ)*(2*(D:ℝ)^2*K+2)^2*N^2 := by |
| let S := (flagSiteBox_finite (a:=a) (m:=m) (v:=v) hD (mul_pos hN hK)).toFinset |
| have hc : ((flagSignalSupport (v:=v) hD g hK hN).card:ℝ) ≤ 2*(S.card:ℝ) := by |
| have hh := (Finset.card_union_le S (S.image (flagSiteAction hD g))).trans |
| (Nat.add_le_add_left (Finset.card_image_le) S.card) |
| change (S∪S.image (flagSiteAction hD g)).card ≤ S.card+S.card at hh |
| have hh' : (((S∪S.image (flagSiteAction hD g)).card:ℝ)) ≤ (S.card:ℝ)+(S.card:ℝ) := by exact_mod_cast hh |
| change ((S∪S.image (flagSiteAction hD g)).card:ℝ) ≤ 2*(S.card:ℝ) |
| simpa only [← two_mul] using hh' |
| have hb := flagSiteBox_card_bound hD (mul_pos hN hK) S |
| (fun z hz => (Set.Finite.mem_toFinset _).mp hz) |
| have hn : 2*(D:ℝ)^2*(N*K)+2 ≤ (2*(D:ℝ)^2*K+2)*N := by nlinarith |
| calc |
| _ ≤ 2*(S.card:ℝ) := hc |
| _ ≤ 2*((m:ℝ)*(2*(D:ℝ)^2*(N*K)+2)^2) := by linarith |
| _ ≤ 2*((m:ℝ)*((2*(D:ℝ)^2*K+2)*N)^2) := by gcongr |
| _ = _ := by ring |
|
|
| theorem flagSignalDifference_energy {a m D : ℕ} {v : ℝ×ℝ} (hD : 0<D) |
| (g : polygonFullGroup a m) (θ : (ℝ×ℝ) → ℝ) {q K L : ℝ} |
| (hK : 0<K) (hcompact : ∀x,K ≤ ‖x‖ → θ x=q) (hL : 0≤L) |
| (hLip : ∀x y,|θ x-θ y| ≤ L*(|x.1-y.1|+|x.2-y.2|)) : |
| ∃C : ℝ,0≤C ∧ ∀N : ℝ,1≤N → ∀T : Finset (FlagSite a m D v), |
| ∑z∈T,(flagSignalDifference hD g θ N z)^2 ≤ C := by |
| obtain ⟨B,hB,hamp⟩ := flagSignalDifference_amplitude (v:=v) hD g θ hL hLip |
| refine ⟨2*(m:ℝ)*(2*(D:ℝ)^2*K+2)^2*B^2,by positivity,fun N hN T => ?_⟩ |
| have hN₀ : 0<N := lt_of_lt_of_le zero_lt_one hN |
| let S := flagSignalSupport (v:=v) hD g hK hN₀ |
| have hsub : ∑z∈T,(flagSignalDifference hD g θ N z)^2 ≤ ∑z∈S,(flagSignalDifference hD g θ N z)^2 := by |
| calc |
| _ = ∑z∈T∩S,(flagSignalDifference hD g θ N z)^2 := by |
| symm |
| apply Finset.sum_subset Finset.inter_subset_left |
| intro z hz hnot |
| have hzS : z∉S := by simpa only [Finset.mem_inter,hz,true_and] using hnot |
| rw [flagSignalDifference_support hD g θ hK hN₀ hcompact z hzS] |
| norm_num |
| _ ≤ _ := Finset.sum_le_sum_of_subset_of_nonneg Finset.inter_subset_right |
| (fun _ _ _ => sq_nonneg _) |
| have hb (z : FlagSite a m D v) : (flagSignalDifference hD g θ N z)^2 ≤ (B/N)^2 := |
| by |
| have hh := (sq_le_sq₀ (abs_nonneg _) (div_nonneg hB hN₀.le)).mpr (hamp N hN₀ z) |
| simpa only [sq_abs] using hh |
| calc |
| _ ≤ ∑z∈S,(flagSignalDifference hD g θ N z)^2 := hsub |
| _ ≤ ∑_z∈S,(B/N)^2 := Finset.sum_le_sum (fun z _ => hb z) |
| _ = (S.card:ℝ)*(B/N)^2 := by simp |
| _ ≤ (2*(m:ℝ)*(2*(D:ℝ)^2*K+2)^2*N^2)*(B/N)^2 := |
| mul_le_mul_of_nonneg_right (flagSignalSupport_card hD g hK hN₀ hN) (sq_nonneg _) |
| _ = _ := by field_simp |
|
|
| end FlagSignalEnergy |
|
|
| end SimpleAmenable |
| end |
| end |
|
|
| end OAI |
|
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