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a44267b | 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 | import OAI.NumberTheory.PrimePackets.ZetaDerivatives
import OAI.NumberTheory.DukePrimeDegree.Geometry.Convergence
namespace OAI
/-!
# Conditional prime-degree packet equidistribution
Complete volume-weighted local-homothety packets converge to Haar
probability and form a tight family under a residue-relative uniform
zeta scale bound and prime-degree positive-entropy rigidity. The
arithmetic and dynamical hypotheses concern the actual homogeneous
lattice space and its diagonal action. A subsequential packet limit
supplies Haar probability.
-/
open MeasureTheory ProbabilityTheory Filter Set
open scoped ENNReal Topology
namespace DukePrimeDegree
open PrimeDegreePackets
universe u
/-- The published rigidity input, on the actual lattice space and with
actual diagonal ergodicity and measure-theoretic entropy. -/
def EKLPrimeInput (d : ℕ) : Prop :=
Nat.Prime (d + 1) → 3 ≤ d + 1 →
∀ ν : Measure (X (d + 1)), IsProbabilityMeasure ν →
ErgodicSMul (Multiplicative (LogCoordinates d)) (X (d + 1)) ν →
0 < measureEntropy ν
(fun x => rightAction x (diagonalOneParameter (regularDiagonalWeights d)
(regularDiagonalWeights_sum d) 1)) → IsHaarProbability ν
/-- Every weak packet limit is Haar under the analytic scale bound and
published prime-degree rigidity. -/
theorem packet_limit_isHaar_of_scale_and_EKL
(d : ℕ) (hprime : Nat.Prime (d + 1)) (hthree : 3 ≤ d + 1) (hodd : Odd (d + 1))
(Hscale : UniformZetaScaleBound.{u} (d + 1)) (ekl : EKLPrimeInput d)
(K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)]
[∀ i, NumberField.IsTotallyReal (K i)]
(M : ∀ i, PrimeDegreePackets.FullLattice (d + 1) (K i))
(σ : ∀ i, OrderedEmbeddings (d + 1) (K i))
(hn : ∀ i, Module.finrank ℚ (K i) = d + 1)
(hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop)
(ν : ProbabilityMeasure (X (d + 1)))
(hlim : Tendsto (fun i => packetProbability (M i) (σ i) (hn i) hodd)
atTop (𝓝 ν)) : IsHaarProbability (ν : Measure (X (d + 1))) := by
obtain ⟨m, hm, hdec, hcomp⟩ :=
packet_limit_positive_ergodic_components_of_zeta_scale d hodd Hscale K M σ hn hD ν hlim
let : MeasurableSpace (X (d + 1)) := xMeasurableSpace (d + 1)
let : @IsProbabilityMeasure (X (d + 1)) m
((ν : Measure (X (d + 1))).trim hm) := by
constructor
rw [trim_measurableSet_eq hm (@MeasurableSet.univ _ m)]
exact measure_univ
have hHaar : ∀ᵐ z ∂(ν : Measure (X (d + 1))).trim hm,
IsHaarProbability (condExpKernel (ν : Measure (X (d + 1))) m z) := by
filter_upwards [hcomp] with z hz
exact ekl hprime hthree _ inferInstance hz.1 hz.2
obtain ⟨z, hz⟩ := hHaar.exists
have heq := measure_eq_haar_of_condExpKernel_invariance
(ν : Measure (X (d + 1)))
(condExpKernel (ν : Measure (X (d + 1))) m z) hz m hm hdec
(hHaar.mono fun _ h => h.2)
exact heq.symm ▸ hz
/-- The full volume-weighted packet conclusion, including no escape of
mass, conditional on the explicitly separated analytic and published
rigidity inputs. The family may vary through arbitrary full lattices and
arbitrary local homothety types. -/
theorem prime_degree_packet_equidistribution_of_scale_and_EKL
(d : ℕ) (hprime : Nat.Prime (d + 1)) (hfive : 5 ≤ d + 1)
(Hscale : UniformZetaScaleBound.{u} (d + 1)) (ekl : EKLPrimeInput d)
(K : ℕ → Type u) [∀ i, Field (K i)] [∀ i, NumberField (K i)]
[∀ i, NumberField.IsTotallyReal (K i)]
(M : ∀ i, PrimeDegreePackets.FullLattice (d + 1) (K i))
(σ : ∀ i, OrderedEmbeddings (d + 1) (K i))
(hn : ∀ i, Module.finrank ℚ (K i) = d + 1)
(hD : Tendsto (fun i => PrimeDegreePackets.multiplierDiscriminant (M i)) atTop atTop) :
∃ ν : ProbabilityMeasure (X (d + 1)),
IsHaarProbability (ν : Measure (X (d + 1))) ∧
Tendsto (fun i => packetProbability (M i) (σ i) (hn i)
(hprime.odd_of_ne_two (by omega))) atTop (𝓝 ν) ∧
IsTightMeasureSet (Set.range (fun i => packetMeasure (M i) (σ i))) := by
let hodd : Odd (d + 1) := hprime.odd_of_ne_two (by omega)
let p : ℕ → ProbabilityMeasure (X (d + 1)) :=
fun i => packetProbability (M i) (σ i) (hn i) hodd
have hQ := packetScale_tendsto_of_multiplierDiscriminant_tendsto
(Nat.succ_pos d) K M hn hD
have htight := packet_tight_of_zeta_scale d hodd Hscale K M σ hn hQ
obtain ⟨ν, φ, hφ, hsub⟩ :=
packet_probability_subsequence_of_zeta_scale d hodd Hscale K M σ hn hD
have hHaar := packet_limit_isHaar_of_scale_and_EKL d hprime (by omega) hodd Hscale ekl
(fun i => K (φ i)) (fun i => M (φ i)) (fun i => σ (φ i))
(fun i => hn (φ i)) (hD.comp hφ.tendsto_atTop) ν hsub
refine ⟨ν, hHaar, ?_, htight⟩
have htight' : IsTightMeasureSet {μ : Measure (X (d + 1)) | ∃ i, μ = p i} := by
convert htight using 1
ext μ
simp only [Set.mem_ofPred_eq, Set.mem_range]
constructor <;> rintro ⟨i, hi⟩ <;> exact ⟨i, hi.symm⟩
apply tendsto_of_tight_of_unique_subsequential_limit p ν htight'
intro ψ hψ ξ hξ
have hHaarξ := packet_limit_isHaar_of_scale_and_EKL d hprime (by omega) hodd Hscale ekl
(fun i => K (ψ i)) (fun i => M (ψ i)) (fun i => σ (ψ i))
(fun i => hn (ψ i)) (hD.comp hψ.tendsto_atTop) ξ hξ
exact Subtype.ext (isHaarProbability_unique hHaarξ hHaar)
end DukePrimeDegree
end OAI
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