| import OAI.NumberTheory.DukePrimeDegree.Arithmetic.StarkZeros |
| import OAI.NumberTheory.DukePrimeDegree.MainConditional |
|
|
| namespace OAI |
|
|
| /-! |
| # Packet equidistribution from exceptional-zero exclusion |
|
|
| Completed-zeta continuation, the zero sum and complex zero separation |
| reduce the analytic hypothesis to real exceptional-zero exclusion. |
| Together with prime-degree positive-entropy rigidity, this gives |
| convergence of complete packets and tightness. |
| -/ |
|
|
| open MeasureTheory Filter |
| open scoped Topology |
|
|
| namespace DukePrimeDegree |
|
|
| open PrimeDegreePackets |
|
|
| universe u |
|
|
| / |
| Only real exceptional-zero exclusion and measure rigidity remain. -/ |
| theorem prime_degree_packet_equidistribution_of_real_zero_exclusion |
| (d : ℕ) (hprime : Nat.Prime (d + 1)) (hfive : 5 ≤ d + 1) |
| (stark : ExceptionalRealZeroExclusion.{u} d) (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 |
| exact prime_degree_packet_equidistribution_of_scale_and_EKL d hprime hfive |
| (uniformZetaScaleBound_of_completedXiGap d |
| (completedXiGap_of_exceptionalRealZeroExclusion d stark)) ekl K M σ hn hD |
|
|
| end DukePrimeDegree |
|
|
| end OAI |
|
|
