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59d1c80 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import OAI.Geometry.LatticeCovering.GaussianPositions
namespace OAI
/-! 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. -/
noncomputable section
namespace SingleLatticeCovering.Assembly
open PrimeKernel SimplexYoung MeasureTheory Folded LatticeGeometry Completion Shear CoverGeometry Vertical Sections Horizontal Inputs
open Filter Topology
open scoped BigOperators Pointwise ENNReal
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
end SingleLatticeCovering.Assembly
end
end OAI
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