| import Mathlib.Probability.Kernel.Composition.Prod |
| import OAI.NumberTheory.Jacobsthal.Paths.HighPrefixState |
|
|
| namespace OAI |
|
|
| namespace Erdos970 |
|
|
| section |
|
|
| namespace ErdosPrimeInputs.HighPrefixKernel |
|
|
| open Set Filter MeasureTheory ProbabilityTheory |
| open scoped ENNReal ProbabilityTheory |
| open WeightedKernelMass HarmonicPrefixKernel HarmonicPrefixTilt HarmonicLongPrefixes HighPrefixState |
|
|
| noncomputable def kernel (ell S : ℝ) : Kernel State State := |
| ((Kernel.id : Kernel State State).prod |
| ((HarmonicPrefixKernel.kernel ell).comap (fun z : State => z.1) measurable_fst)).map |
| (Function.uncurry (update S)) |
| |
| instance kernel_isSFinite (ell S : ℝ) : IsSFiniteKernel (kernel ell S) := by |
| unfold kernel |
| infer_instance |
| |
| lemma update_at_measurable (S : ℝ) (z : State) : Measurable (update S z) := |
| (update_measurable S).of_uncurry_left |
| |
| theorem kernel_eq_map (ell S : ℝ) (z : State) : |
| kernel ell S z = (HarmonicPrefixKernel.kernel ell z.1).map (update S z) := by |
| apply Measure.ext_of_lintegral |
| intro f hf |
| rw [kernel,Kernel.map_apply _ (update_measurable S),lintegral_map hf (update_measurable S)] |
| have hg : Measurable (fun p : State × ℝ => f (update S p.1 p.2)) := hf.comp (update_measurable S) |
| calc |
| _ = ∫⁻ t, f (update S z t) ∂HarmonicPrefixKernel.kernel ell z.1 := |
| Kernel.lintegral_id_prod (f := fun p : State × ℝ => f (update S p.1 p.2)) hg _ z |
| _ = _ := (lintegral_map hf (update_at_measurable S z)).symm |
| |
| lemma harmonic_ae_interval (ell B : ℝ) : |
| ∀ᵐ t ∂HarmonicPrefixKernel.kernel ell B, t ∈ Ioc ell B := by |
| rw [HarmonicPrefixKernel.kernel_eq_restrict, |
| ae_withDensity_iff (by fun_prop : Measurable (fun t : ℝ => ENNReal.ofReal (t⁻¹)))] |
| filter_upwards [self_mem_ae_restrict measurableSet_Ioc] with t ht |
| intro _ |
| exact ht |
| |
| theorem ae_update (ell S : ℝ) (z : State) {P : State → Prop} |
| (hP : MeasurableSet {w | P w}) (h : ∀ t ∈ Ioc ell z.1, P (update S z t)) : |
| ∀ᵐ w ∂kernel ell S z, P w := by |
| rw [kernel_eq_map] |
| apply (ae_map_iff (update_at_measurable S z).aemeasurable hP).mpr |
| exact (harmonic_ae_interval ell z.1).mono (fun t ht => h t ht) |
| |
| lemma state_schur_one {ell : ℝ} (hell : 0<ell) (S : ℝ) (z : State) : |
| 1+(∫⁻ w, boundE ell w.1 ∂kernel ell S z) = boundE ell z.1 := by |
| rw [kernel_eq_map] |
| have hf : Measurable (fun w : State => boundE ell w.1) := (boundE_measurable ell).comp measurable_fst |
| rw [lintegral_map (f := fun w : State => boundE ell w.1) hf (update_at_measurable S z)] |
| exact schur_one hell z.1 |
| |
| lemma state_schur_two {ell : ℝ} (hell : 0<ell) (S : ℝ) (z : State) : |
| 1+2*(∫⁻ w, squareBoundE ell w.1 ∂kernel ell S z) = squareBoundE ell z.1 := by |
| rw [kernel_eq_map] |
| have hf : Measurable (fun w : State => squareBoundE ell w.1) := (squareBoundE_measurable ell).comp measurable_fst |
| rw [lintegral_map (f := fun w : State => squareBoundE ell w.1) hf (update_at_measurable S z)] |
| exact schur_two hell z.1 |
| |
| theorem total_mass {ell B : ℝ} (hell : 0<ell) (hB : ell≤B) (S R : ℝ) : |
| (∑' n : ℕ, mass (kernel ell S) n (initial S B R)) ≤ ENNReal.ofReal (B/ell) := by |
| have h := weighted_total_bound (kernel ell S) 1 (fun z : State => boundE ell z.1) |
| (fun z => by simpa only [one_mul] using (state_schur_one hell S z).le) (initial S B R) |
| simpa only [one_pow,one_mul,initial,boundE,bound_on hell hB] using h |
| |
| theorem weighted_mass {ell B : ℝ} (hell : 0<ell) (hB : ell≤B) (S R : ℝ) : |
| (∑' n : ℕ, (2:ℝ≥0∞)^n*mass (kernel ell S) n (initial S B R)) ≤ ENNReal.ofReal ((B/ell)^2) := by |
| have h := weighted_total_bound (kernel ell S) 2 (fun z : State => squareBoundE ell z.1) |
| (fun z => (state_schur_two hell S z).le) (initial S B R) |
| simpa only [initial,squareBoundE,bound_on hell hB] using h |
| |
| theorem ae_invariant {ell B : ℝ} (hell : 0<ell) (hB : ell≤B) (S R : ℝ) : |
| ∀ n : ℕ, (n:ℝ)≤S → |
| ∀ᵐ z ∂(kernel ell S ^ n) (initial S B R), Invariant ell S n z := by |
| intro n |
| induction n with |
| | zero => |
| intro _ |
| change ∀ᵐ z ∂Measure.dirac (initial S B R), Invariant ell S 0 z |
| exact (ae_dirac_iff (invariant_measurable ell S 0)).mpr (initial_invariant hell hB) |
| | succ n ih => |
| intro hn |
| have hn' : (n:ℝ)≤S := by push_cast at hn; linarith |
| rw [pow_succ'] |
| apply Kernel.ae_comp_of_ae_ae (invariant_measurable ell S (n+1)) |
| filter_upwards [ih hn'] with z hz |
| exact ae_update ell S z (invariant_measurable ell S (n+1)) |
| (fun t ht => update_invariant hell n z t hz ht hn') |
| |
| end ErdosPrimeInputs.HighPrefixKernel |
| |
| end |
| |
| end Erdos970 |
| |
| end OAI |
| |
