| import Mathlib |
| import OAI.GroupTheory.SimpleAmenable.RandomFields.WeightedAveragingError |
|
|
| namespace OAI |
|
|
| section |
| section |
| open scoped symmDiff |
| namespace SimpleAmenable |
| open scoped commutatorElement |
| open scoped commutatorElement |
| section SignalCutoff |
| open Classical |
|
|
| theorem scaledSiteConjugate_displacement {a m D : ℕ} {v : ℝ×ℝ} (hD : 0<D) |
| (g : polygonFullGroup a m) : |
| ∃Q : ℝ,0 ≤ Q ∧ ∀N : ℝ,0<N → ∀z : FlagSite a m D v, |
| ‖scaledSiteConjugate N (flagSiteAction hD g z)-scaledSiteConjugate N z‖ ≤ Q/N := by |
| obtain ⟨Q,hQ,hb⟩ := flagSiteAction_displacement (v:=v) hD g |
| refine ⟨Q,hQ,fun N hN z => ?_⟩ |
| rw [Prod.norm_def] |
| apply max_le <;> simp only [scaledSiteConjugate,Prod.fst_sub,Prod.snd_sub,Real.norm_eq_abs, |
| ← sub_div,abs_div,abs_of_pos hN] |
| · apply div_le_div_of_nonneg_right _ hN.le |
| exact (le_add_of_nonneg_right (abs_nonneg _)).trans (hb z) |
| · apply div_le_div_of_nonneg_right _ hN.le |
| exact (le_add_of_nonneg_left (abs_nonneg _)).trans (hb z) |
|
|
| theorem flagSignalDifference_ne_zero_near {a m D : ℕ} {v : ℝ×ℝ} (hD : 0<D) |
| (g : polygonFullGroup a m) (ψ : (ℝ×ℝ) → ℝ) {K Q N : ℝ} (q : ℝ) |
| (hcompact : ∀x,K ≤ ‖x‖ → ψ x=0) |
| (hQ : 0 ≤ Q) (hN : 0<N) |
| (hdisp : ∀z : FlagSite a m D v, |
| ‖scaledSiteConjugate N (flagSiteAction hD g⁻¹ z)-scaledSiteConjugate N z‖ ≤ Q/N) |
| (z : FlagSite a m D v) (hz : flagSignalDifference hD g (fun x => q+ψ x) N z≠0) : |
| ‖scaledSiteConjugate N z‖ < K+Q/N := by |
| rw [flagSignalDifference_add_constant] at hz |
| by_cases hh : ψ (scaledSiteConjugate N z)=0 |
| · have hi : ψ (scaledSiteConjugate N (flagSiteAction hD g⁻¹ z))≠0 := by |
| intro he |
| exact hz (by simp [flagSignalDifference,flagMeanSignal,he,hh]) |
| have hn : ‖scaledSiteConjugate N (flagSiteAction hD g⁻¹ z)‖<K := |
| lt_of_not_ge (fun he => hi (hcompact _ he)) |
| have ht := norm_le_norm_add_norm_sub (scaledSiteConjugate N (flagSiteAction hD g⁻¹ z)) |
| (scaledSiteConjugate N z) |
| linarith [hdisp z] |
| · have hn : ‖scaledSiteConjugate N z‖<K := lt_of_not_ge (fun he => hh (hcompact _ he)) |
| exact hn.trans_le (le_add_of_nonneg_right (div_nonneg hQ hN.le)) |
|
|
| theorem flagSignalDifference_cutoff {a m D : ℕ} {v : ℝ×ℝ} (hD : 0<D) |
| (g : polygonFullGroup a m) (ψ χ : (ℝ×ℝ) → ℝ) {K : ℝ} (q : ℝ) |
| (hcompact : ∀x,K ≤ ‖x‖ → ψ x=0) |
| (hχ : ∀x,‖x‖ ≤ K+2 → χ x=1) : |
| ∃Q : ℝ,0 ≤ Q ∧ ∀(N r η : ℝ),0<N → Q ≤ N → 0<η → η ≤ 1 → |
| ∀z w : FlagSite a m D v,flagRowTent (r/N) (N*η) z w≠0 → |
| flagSignalDifference hD g (fun x => q+ψ x) N w≠0 → |
| χ (scaledSiteConjugate N z)=1 ∧ χ (scaledSiteConjugate N w)=1 := by |
| obtain ⟨Q,hQ,hdisp⟩ := scaledSiteConjugate_displacement (v:=v) hD g⁻¹ |
| refine ⟨Q,hQ,fun N r η hN hQN hη hη₁ z w hkw hw => ?_⟩ |
| have hwn := flagSignalDifference_ne_zero_near hD g ψ q hcompact hQ hN (hdisp N hN) w hw |
| have hQN' : Q/N ≤ 1 := (div_le_one hN).mpr hQN |
| have hn := planeTent_scaled_near hN hη z w |
| ((flagRowTent_ne_zero_iff _ _ _ _).mp hkw).2.2 |
| have hz := norm_le_norm_add_norm_sub (scaledSiteConjugate N w) (scaledSiteConjugate N z) |
| rw [norm_sub_rev] at hz |
| exact ⟨hχ _ (by linarith),hχ _ (by linarith)⟩ |
|
|
| theorem compactSignal_amplitude {a m D : ℕ} {v : ℝ×ℝ} (hD : 0<D) |
| (g : polygonFullGroup a m) (ψ : (ℝ×ℝ) → ℝ) |
| (hψ : ContDiff ℝ 2 ψ) (hs : HasCompactSupport ψ) (q : ℝ) : |
| ∃C : ℝ,0 ≤ C ∧ ∀N : ℝ,0<N → ∀z : FlagSite a m D v, |
| |flagSignalDifference hD g (fun x => q+ψ x) N z| ≤ C/N := by |
| obtain ⟨L₀,L₁,hL₀,hL₁⟩ := compactSignal_uniform_constants ψ hψ hs |
| obtain ⟨C,hC,hamp⟩ := flagSignalDifference_amplitude (v:=v) hD g ψ L₀.coe_nonneg (by |
| intro x y |
| have hh := hL₀.dist_le_mul x y |
| rw [Real.dist_eq,dist_eq_norm,Prod.norm_def] at hh |
| refine hh.trans (mul_le_mul_of_nonneg_left ?_ L₀.coe_nonneg) |
| simp only [Prod.fst_sub,Prod.snd_sub,Real.norm_eq_abs] |
| exact max_le (le_add_of_nonneg_right (abs_nonneg _)) (le_add_of_nonneg_left (abs_nonneg _))) |
| refine ⟨C,hC,fun N hN z => ?_⟩ |
| rw [flagSignalDifference_add_constant] |
| exact hamp N hN z |
|
|
| end SignalCutoff |
|
|
| section AveragingL1Rows |
| open Classical |
|
|
| theorem flagAveragingMatrix_row_sum {a m D : ℕ} {v : ℝ×ℝ} (hD : 0<D) |
| (χ : (ℝ×ℝ) → ℝ) (hχ : ∀x,χ x∈Set.Icc (0:ℝ) 1) |
| {ρ N r η κ : ℝ} (hρ : 0<ρ) (hN : 0<N) (hr : 0<r) (hη : 0<η) (hκ : 0<κ) |
| (z : FlagSite a m D v) (S : Finset (FlagSite a m D v)) : |
| ∑w∈S,flagAveragingMatrix χ ρ N r η κ z w ≤ |
| (4*(D:ℝ)^2*r*η+2)^2/(ρ*(r*η)^2) := by |
| let T := S.filter (fun w => flagAveragingMatrix χ ρ N r η κ z w≠0) |
| have he : (∑w∈S,flagAveragingMatrix χ ρ N r η κ z w)= |
| ∑w∈T,flagAveragingMatrix χ ρ N r η κ z w := by |
| symm |
| apply Finset.sum_subset (Finset.filter_subset _ _) |
| intro w hw hwt |
| simpa [T,hw] using hwt |
| rw [he] |
| have hc := flagAveragingMatrix_row_card hD χ ρ κ hN hr hη z T |
| (fun w hw => (Finset.mem_filter.mp hw).2) |
| calc |
| _ ≤ ∑_w∈T,(ρ*(r*η)^2)⁻¹ := Finset.sum_le_sum |
| (fun w _ => (flagAveragingMatrix_entry_bounds χ hχ hρ hN hr hη hκ z w).2) |
| _ = (T.card:ℝ)*(ρ*(r*η)^2)⁻¹ := by simp |
| _ ≤ (4*(D:ℝ)^2*r*η+2)^2*(ρ*(r*η)^2)⁻¹ := mul_le_mul_of_nonneg_right hc (by positivity) |
| _ = _ := by rw [div_eq_mul_inv] |
|
|
| theorem flagAveragingMatrix_row_sum_uniform {a m D : ℕ} {v : ℝ×ℝ} (hD : 0<D) |
| (χ : (ℝ×ℝ) → ℝ) (hχ : ∀x,χ x∈Set.Icc (0:ℝ) 1) |
| {ρ N r η κ : ℝ} (hρ : 0<ρ) (hN : 0<N) (hr : 0<r) (hη : 0<η) (hκ : 0<κ) |
| (hscale : 1 ≤ r*η) (z : FlagSite a m D v) (S : Finset (FlagSite a m D v)) : |
| ∑w∈S,flagAveragingMatrix χ ρ N r η κ z w ≤ (4*(D:ℝ)^2+2)^2/ρ := by |
| have hh := flagAveragingMatrix_row_sum hD χ hχ hρ hN hr hη hκ z S |
| refine hh.trans ?_ |
| have hle : 4*(D:ℝ)^2*r*η+2 ≤ (4*(D:ℝ)^2+2)*(r*η) := by nlinarith |
| have hb := pow_le_pow_left₀ (show 0 ≤ 4*(D:ℝ)^2*r*η+2 by positivity) hle 2 |
| calc |
| _ ≤ ((4*(D:ℝ)^2+2)*(r*η))^2/(ρ*(r*η)^2) := div_le_div_of_nonneg_right hb (by positivity) |
| _ = _ := by field_simp |
|
|
| theorem flagAveragingMatrix_crude_error {a m D : ℕ} {v : ℝ×ℝ} (hD : 0<D) |
| (χ : (ℝ×ℝ) → ℝ) (hχ : ∀x,χ x∈Set.Icc (0:ℝ) 1) |
| (h : FlagSite a m D v → ℝ) {ρ N r η κ H : ℝ} |
| (hρ : 0<ρ) (hN : 0<N) (hr : 0<r) (hη : 0<η) (hκ : 0<κ) |
| (hscale : 1 ≤ r*η) (hH : 0 ≤ H) (hamp : ∀z,|h z| ≤ H) |
| (z : FlagSite a m D v) (S : Finset (FlagSite a m D v)) : |
| |(∑w∈S,flagAveragingMatrix χ ρ N r η κ z w*h w)-h z| ≤ |
| ((4*(D:ℝ)^2+2)^2/ρ+1)*H := by |
| have hsum : |∑w∈S,flagAveragingMatrix χ ρ N r η κ z w*h w| ≤ |
| ((4*(D:ℝ)^2+2)^2/ρ)*H := by |
| calc |
| _ ≤ ∑w∈S,|flagAveragingMatrix χ ρ N r η κ z w*h w| := Finset.abs_sum_le_sum_abs _ _ |
| _ ≤ ∑w∈S,flagAveragingMatrix χ ρ N r η κ z w*H := by |
| apply Finset.sum_le_sum |
| intro w hw |
| have hm := (flagAveragingMatrix_entry_bounds χ hχ hρ hN hr hη hκ z w).1 |
| rw [abs_mul,abs_of_nonneg hm] |
| exact mul_le_mul_of_nonneg_left (hamp w) hm |
| _ = (∑w∈S,flagAveragingMatrix χ ρ N r η κ z w)*H := (Finset.sum_mul _ _ _).symm |
| _ ≤ _ := mul_le_mul_of_nonneg_right |
| (flagAveragingMatrix_row_sum_uniform hD χ hχ hρ hN hr hη hκ hscale z S) hH |
| have ht := abs_sub (∑w∈S,flagAveragingMatrix χ ρ N r η κ z w*h w) (h z) |
| nlinarith [hamp z] |
|
|
| end AveragingL1Rows |
|
|
| end SimpleAmenable |
| end |
| end |
|
|
| end OAI |
|
|
