| import OAI.NumberTheory.GaussianMoat.SignedArithmetic |
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| namespace OAI |
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| universe uIota |
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| namespace GaussianMoat |
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| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
| noncomputable def fairCoin : Measure Bool := (PMF.uniformOfFintype Bool).toMeasure |
| instance fairCoin_probability : IsProbabilityMeasure fairCoin := by |
| unfold fairCoin |
| infer_instance |
| noncomputable def signMeasure (ι : Type uIota) [Fintype ι] : Measure (ι → Bool) := |
| Measure.pi (fun _ => fairCoin) |
| instance signMeasure_probability (ι : Type uIota) [Fintype ι] : IsProbabilityMeasure (signMeasure ι) := by |
| unfold signMeasure |
| infer_instance |
| |
| theorem fairCoin_mean (f : Bool → ℝ) : ∫ b, f b ∂fairCoin = (f false + f true) / 2 := by |
| have hf : Integrable f fairCoin := Integrable.of_mem_Icc |
| (min (f false) (f true)) (max (f false) (f true)) (by fun_prop) (by |
| apply ae_of_all |
| intro b |
| cases b <;> simp) |
| rw [integral_fintype hf] |
| simp [fairCoin, measureReal_def, |
| PMF.uniformOfFintype_apply] |
| ring |
| |
| theorem sign_hoeffding {ι : Type uIota} [Fintype ι] (f : ι → Bool → ℝ) (w : ι → ℝ) |
| (hf : ∀ i b, f i b ∈ Set.Icc 0 (w i)) {ε : ℝ} (hε : 0 ≤ ε) : |
| (signMeasure ι).real {σ | ε + ∑ i, (f i false + f i true) / 2 ≤ ∑ i, f i (σ i)} |
| ≤ Real.exp (-ε ^ 2 / (2 * ∑ i, (((‖w i‖₊ / 2) ^ 2 : ℝ≥0) : ℝ))) := by |
| classical |
| let X : ι → (ι → Bool) → ℝ := fun i σ => f i (σ i) - (f i false + f i true) / 2 |
| have hsub (i : ι) : HasSubgaussianMGF (X i) ((‖w i‖₊ / 2) ^ 2) (signMeasure ι) := by |
| have h := hasSubgaussianMGF_of_mem_Icc (μ := fairCoin) (X := f i) |
| (by fun_prop) (ae_of_all _ (hf i)) |
| rw [fairCoin_mean] at h |
| simp only [sub_zero] at h |
| have he := (measurePreserving_eval (fun _ : ι => fairCoin) i).map_eq |
| rw [← he] at h |
| have heval : AEMeasurable (Function.eval i : (ι → Bool) → Bool) (signMeasure ι) := |
| (measurable_pi_apply i).aemeasurable |
| exact HasSubgaussianMGF.of_map heval h |
| have hind : iIndepFun X (signMeasure ι) := |
| iIndepFun_pi (X := fun i b => f i b - (f i false + f i true) / 2) (fun _ => by fun_prop) |
| have h := HasSubgaussianMGF.measure_sum_ge_le_of_iIndepFun hind |
| (s := Finset.univ) (fun i _ => hsub i) hε |
| simpa only [X, Finset.sum_sub_distrib, le_sub_iff_add_le, NNReal.coe_sum] using h |
| |
| open scoped Classical in |
| |
| noncomputable def SplitSieve.eligible (s : SplitSieve) (v : GaussianInt) : Finset ℕ := |
| s.primes.filter (fun p => s.factor p ∣ v ∨ star (s.factor p) ∣ v) |
| |
| noncomputable def SplitSieve.lineLog (s : SplitSieve) (v : GaussianInt) (p : ℕ) (b : Bool) : ℝ := by |
| classical |
| exact if signedFactor (s.factor p) b ∣ v then Real.log p else 0 |
| |
| theorem SplitSieve.eligible_log_bound (s : SplitSieve) {v : GaussianInt} (hv : v ≠ 0) : |
| ∑ p ∈ s.eligible v, Real.log (p : ℝ) ≤ Real.log (v.norm : ℝ) := by |
| classical |
| let σ : ℕ → Bool := fun p => decide (s.factor p ∣ v) |
| have hnorm := s.eligible_product_le_norm (s.eligible v) |
| (Finset.filter_subset _ _) σ hv (by |
| intro p hp |
| obtain ⟨_, hleft | hright⟩ := Finset.mem_filter.mp hp |
| · simp [σ, signedFactor, hleft] |
| · by_cases hd : s.factor p ∣ v |
| · simp [σ, signedFactor, hd] |
| · simpa [σ, signedFactor, hd] using hright) |
| have hnorm' : ∏ p ∈ s.eligible v, (p : ℝ) ≤ (v.norm : ℝ) := by |
| have h : ((∏ p ∈ s.eligible v, (p : ℤ)) : ℝ) ≤ (v.norm : ℝ) := by exact_mod_cast hnorm |
| push_cast at h |
| exact h |
| have hpos (p : ℕ) (hp : p ∈ s.eligible v) : 0 < (p : ℝ) := by |
| exact_mod_cast (s.prime_mem p (Finset.mem_filter.mp hp).1).pos |
| have hlog := Real.log_le_log (Finset.prod_pos hpos) hnorm' |
| rw [Real.log_prod (fun p hp => (hpos p hp).ne')] at hlog |
| exact hlog |
| |
| theorem SplitSieve.lineLog_mean (s : SplitSieve) {v : GaussianInt} |
| (hv : PrimitiveDirection v) (hv0 : v ≠ 0) : |
| ∑ p : s.primes, (s.lineLog v p false + s.lineLog v p true) / 2 ≤ |
| Real.log (v.norm : ℝ) / 2 := by |
| classical |
| have heq : ∑ p : s.primes, (s.lineLog v p false + s.lineLog v p true) = |
| ∑ p ∈ s.eligible v, Real.log (p : ℝ) := by |
| rw [Finset.sum_coe_sort s.primes (fun p => s.lineLog v p false + s.lineLog v p true), |
| SplitSieve.eligible, Finset.sum_filter] |
| apply Finset.sum_congr rfl |
| intro p hp |
| have hp2 : p ≠ 2 := by have hm := s.mod_four p hp; omega |
| have hnot := not_both_factors_dvd_primitive (s.prime_mem p hp) hp2 (s.factor_norm p hp) hv |
| by_cases h1 : s.factor p ∣ v <;> by_cases h2 : star (s.factor p) ∣ v |
| · exact False.elim (hnot ⟨h1,h2⟩) |
| all_goals simp [SplitSieve.lineLog, signedFactor, h1, h2] |
| rw [← Finset.sum_div, heq] |
| exact div_le_div_of_nonneg_right (s.eligible_log_bound hv0) (by norm_num) |
| |
| theorem SplitSieve.lineLog_tail (s : SplitSieve) {v : GaussianInt} |
| (hv : PrimitiveDirection v) (hv0 : v ≠ 0) {ε : ℝ} (hε : 0 ≤ ε) : |
| (signMeasure s.primes).real {σ | |
| Real.log (v.norm : ℝ) / 2 + ε ≤ ∑ p : s.primes, s.lineLog v p (σ p)} |
| ≤ Real.exp (-ε ^ 2 / (2 * ∑ p : s.primes, |
| (((‖Real.log (p : ℝ)‖₊ / 2) ^ 2 : ℝ≥0) : ℝ))) := by |
| classical |
| apply le_trans (measureReal_mono (s₂ := {σ | |
| ε + ∑ p : s.primes, (s.lineLog v p false + s.lineLog v p true) / 2 ≤ |
| ∑ p : s.primes, s.lineLog v p (σ p)}) ?_) |
| · apply sign_hoeffding (fun p : s.primes => s.lineLog v p) (fun p => Real.log (p : ℝ)) |
| · intro p b |
| have hp : 0 ≤ Real.log (p : ℝ) := Real.log_nonneg (by |
| exact_mod_cast (s.prime_mem p p.property).one_lt.le) |
| unfold SplitSieve.lineLog |
| split_ifs |
| · exact ⟨hp, le_rfl⟩ |
| · exact ⟨le_rfl, hp⟩ |
| · exact hε |
| · intro σ hσ |
| have hmean := s.lineLog_mean hv hv0 |
| change Real.log (v.norm : ℝ) / 2 + ε ≤ _ at hσ |
| change ε + _ ≤ _ |
| linarith |
| |
| end GaussianMoat |
| |
| end OAI |
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