| import OAI.NumberTheory.DukePrimeDegree.MainStarkDischarged |
| import OAI.NumberTheory.DukePrimeDegree.Entropy.RegularEntropyDischarge |
|
|
| namespace OAI |
|
|
| /-! # Prime-degree torus-packet equidistribution |
|
|
| The manuscript's full packet theorem, after discharging both the Stark |
| zero-exclusion input and the prime-degree positive-entropy rigidity input. |
| The lattice and field may vary, so nonmaximal multiplier orders and |
| arbitrary prescribed local homothety types are included. Tightness gives |
| no escape of mass for the complete volume-weighted packets. |
| -/ |
|
|
| open MeasureTheory Filter |
| open scoped Topology |
| namespace DukePrimeDegree |
| open PrimeDegreePackets |
|
|
| universe u |
|
|
| / |
| probability as their multiplier-order discriminants tend to infinity. -/ |
| theorem prime_degree_packet_equidistribution_unconditional |
| (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1) |
| (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 |
| exact prime_degree_packet_equidistribution_of_EKL d hprime hfive |
| (eklPrimeInput_unconditional d) K M σ hn hD |
|
|
| / |
| theorem prime_degree_packets_tendsto_haar_unconditional |
| (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1) |
| (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) |
| (m : ProbabilityMeasure (X (d+1))) |
| (hm : IsHaarProbability (m : Measure (X (d+1)))) : |
| Tendsto (fun i => packetProbability (M i) (σ i) (hn i) |
| (hprime.odd_of_ne_two (by omega))) atTop (𝓝 m) := by |
| obtain ⟨ν,hν,hconv,_⟩ := |
| prime_degree_packet_equidistribution_unconditional d hprime hfive K M σ hn hD |
| have heq : ν=m := Subtype.ext (isHaarProbability_unique hν hm) |
| rwa [heq] at hconv |
|
|
| / |
| including tightness of the entire family. -/ |
| theorem prime_degree_packet_measure_equidistribution_unconditional |
| (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1) |
| (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 : Measure (X (d+1)), IsHaarProbability ν ∧ |
| WeakProbabilityConvergence (fun i => packetMeasure (M i) (σ i)) ν ∧ |
| IsTightMeasureSet (Set.range (fun i => packetMeasure (M i) (σ i))) := by |
| obtain ⟨ν, hν, hconv, htight⟩ := |
| prime_degree_packet_equidistribution_unconditional d hprime hfive K M σ hn hD |
| refine ⟨(ν : Measure (X (d+1))), hν, ?_, htight⟩ |
| exact (weakProbabilityConvergence_iff |
| (fun i => packetProbability (M i) (σ i) (hn i) |
| (hprime.odd_of_ne_two (by omega))) ν).mpr hconv |
|
|
| / |
| probability, in the formulation by bounded continuous test functions. -/ |
| theorem prime_degree_packet_measures_tendsto_haar_unconditional |
| (d : ℕ) (hprime : Nat.Prime (d+1)) (hfive : 5 ≤ d+1) |
| (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) |
| (m : ProbabilityMeasure (X (d+1))) |
| (hm : IsHaarProbability (m : Measure (X (d+1)))) : |
| WeakProbabilityConvergence (fun i => packetMeasure (M i) (σ i)) |
| (m : Measure (X (d+1))) := by |
| exact (weakProbabilityConvergence_iff |
| (fun i => packetProbability (M i) (σ i) (hn i) |
| (hprime.odd_of_ne_two (by omega))) m).mpr |
| (prime_degree_packets_tendsto_haar_unconditional d hprime hfive K M σ hn hD m hm) |
|
|
| end DukePrimeDegree |
|
|
| end OAI |
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|
