| import OAI.Geometry.LatticeCovering.GaussianPositions |
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| namespace OAI |
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| /-! The unconditional eventual covering bound, using the actual prime-congruence |
| and local affine Gaussian-position producers. Conditional intermediate interfaces |
| retain their stated hypotheses and are not used as unproved assumptions here. -/ |
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| noncomputable section |
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| namespace SingleLatticeCovering.Assembly |
| open PrimeKernel SimplexYoung MeasureTheory Folded LatticeGeometry Completion Shear CoverGeometry Vertical Sections Horizontal Inputs |
| open Filter Topology |
| open scoped BigOperators Pointwise ENNReal |
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| theorem eventual_single_lattice_covering : |
| ∃ 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 |
| let CT : ℝ := 301 |
| have hCT : 0 ≤ CT := by norm_num [CT] |
| have hCTeq : 1+CT=302 := by norm_num [CT] |
| 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) hCT |
| refine ⟨Real.exp 1*Cstar,by positivity,?_⟩ |
| filter_upwards [hconditions,FinalRates.eventually_marginal_scales hC (by norm_num : (0:ℝ) < 1) |
| (by norm_num : (0:ℝ) < 1) (by norm_num : (0:ℝ) < 1) (by norm_num : (0:ℝ) ≤ 0), |
| FinalRates.firstSize_tendsto.eventually (eventually_ge_atTop cutoff), |
| (FinalRates.localScale_tendsto hC).eventually GaussianNumerics.eventually_local_coordinate_position, |
| FinalRates.eventually_localScale_root hC, |
| FinalRates.eventually_dimension_small hC,eventually_ge_atTop (540 : ℕ)] |
| with n hn hscale hfirst hgaussian hroot hdim hn540 |
| 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 |
| have hm270 : 270 ≤ n-c.fullDim := by |
| have hd := (hdim c.fullDim hDdim).2 |
| have hnR : (540 : ℝ) ≤ n := by exact_mod_cast hn540 |
| have hh : (270 : ℝ) ≤ ((n-c.fullDim : ℕ) : ℝ) := by |
| rw [Nat.cast_sub hDn] |
| linarith |
| exact_mod_cast hh |
| rw [hCTeq] at hcrit |
| obtain ⟨_,_,_,_,_,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 (FinalRates.dimension_le_localScale hC hDdim) |
| (by omega : 0 < m+c.fullDim) (by simpa only [hmn] using hroot) 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 he y |
| apply hy.trans |
| calc |
| 6*(c.fullDim:ℝ)*R ≤ 6*(FinalRates.localScale C n:ℝ)*(FinalRates.localScale C n:ℝ) := by |
| gcongr |
| · exact (FinalRates.sectionRadius_pos hC n).le |
| · exact_mod_cast FinalRates.dimension_le_localScale hC hDdim |
| · exact FinalRates.sectionRadius_le_localScale C n |
| _ = 6*(FinalRates.localScale C n:ℝ)^2 := by ring |
| obtain ⟨Λ,hd,hfull,hcover,hbound⟩ := sections_and_circuits_cover c hm270 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)) hρ (Real.sqrt_nonneg _) (by positivity) |
| (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) |
| 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 |
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| end SingleLatticeCovering.Assembly |
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| end |
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| end OAI |
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