| import Mathlib |
| import OAI.NumberTheory.CubicGram.PoissonBounds |
|
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| namespace OAI |
|
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| /-! Uniform quantitative recurrence and exponent monotonicity. -/ |
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| section |
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| noncomputable section |
| open scoped BigOperators ContDiff |
| attribute [local instance] Classical.propDecidable |
| namespace CubicFirstMoment |
|
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| def sieveRecurrenceShape (ξ M N : ℝ) : ℝ := |
| M+(M*N)^(2/3 : ℝ)+N+M^(1-ξ)*N^(2*ξ-1) |
|
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| lemma sieveRecurrenceShape_nonneg {ξ M N : ℝ} (hM : 0 ≤ M) (hN : 0 ≤ N) : |
| 0 ≤ sieveRecurrenceShape ξ M N := by unfold sieveRecurrenceShape; positivity |
|
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| lemma scaled_poisson_error_le {ξ M N m k : ℝ} (hξ : 1 ≤ ξ) |
| (hM : 0 < M) (hN : 0 < N) (hm : 1 ≤ m) (hk : m ≤ k) : |
| (M/m)^(1-ξ)*(N/k)^(2*ξ-1) ≤ M^(1-ξ)*N^(2*ξ-1) := by |
| have hm0 : 0 < m := zero_lt_one.trans_le hm |
| have hk0 : 0 < k := hm0.trans_le hk |
| have hk1 : 1 ≤ k := hm.trans hk |
| have hp : m^(ξ-1) ≤ k^(2*ξ-1) := by |
| calc |
| _ ≤ k^(ξ-1) := Real.rpow_le_rpow hm0.le hk (by linarith) |
| _ ≤ _ := Real.rpow_le_rpow_of_exponent_le hk1 (by linarith) |
| have hden : 1 ≤ m^(1-ξ)*k^(2*ξ-1) := by |
| rw [show (1 : ℝ)-ξ = -(ξ-1) by ring,Real.rpow_neg hm0.le] |
| rw [inv_mul_eq_div] |
| exact (one_le_div (by positivity)).mpr hp |
| rw [Real.div_rpow hM.le hm0.le,Real.div_rpow hN.le hk0.le,← mul_div_mul_comm] |
| exact div_le_self (by positivity) hden |
|
|
| lemma SieveExponent.common_cost_bound {ξ : ℝ} (hξ : 1 ≤ ξ) (hξ2 : ξ ≤ 2) |
| (hExp : SieveExponent ξ) {δ : ℝ} (hδ : 0 < δ) (hδ1 : δ ≤ 1) : |
| ∃ C : ℝ, 0 < C ∧ ∀ (S : Finset Eisenstein) (M N : ℝ), 1 ≤ N → N ≤ M → |
| (∀ a ∈ S, primary a ∧ Squarefree a ∧ N ≤ norm a ∧ norm a ≤ Real.sqrt 2*N) → |
| ∀ k ∈ commonRowFactors S, ∀ s ∈ (primaryPrimeFactors k).powerset, |
| commonBlockCost S cubicSieveCutoff M N 8 k (∏ p ∈ s, p) ≤ |
| C*(M*N)^δ*sieveRecurrenceShape ξ M N := by |
| obtain ⟨D,hD,hdual⟩ := hExp.dual_bound hξ hξ2 hδ hδ1 |
| obtain ⟨E,hE,hdiag⟩ := primaryCharacterGram_one_one_bound cubicSieveCutoff |
| cubicSieveCutoff_compact cubicSieveCutoff_smooth.continuous |
| let C : ℝ := 54000*D+E |
| have hC : 0 < C := by dsimp [C]; positivity |
| have hDC : 54000*D ≤ C := by dsimp [C]; linarith |
| have hEC : E ≤ C := by dsimp [C]; linarith |
| refine ⟨C,hC,?_⟩ |
| intro S M N hN hNM hS k hkS s hs |
| let m := ∏ p ∈ s, p |
| have hS' : ∀ a ∈ S, primary a ∧ Squarefree a := fun a ha => ⟨(hS a ha).1,(hS a ha).2.1⟩ |
| have hk := commonRowFactors_spec hS' hkS |
| have hm : primary m := primary_finset_prod _ _ |
| (fun p hp => (primaryPrimeFactor_spec hk.1 (Finset.mem_powerset.mp hs hp)).1.1) |
| have hmk : m ∣ k := (primary_subsets_prod_dvd hk.1 (Finset.mem_powerset.mp hs) k).mpr |
| (fun p hp => (primaryPrimeFactor_spec hk.1 (Finset.mem_powerset.mp hs hp)).2) |
| have hnk1 : 1 ≤ norm k := one_le_norm (primary_ne_zero hk.1) |
| have hnm1 : 1 ≤ norm m := one_le_norm (primary_ne_zero hm) |
| have hnk : 0 < norm k := zero_lt_one.trans_le hnk1 |
| have hnm : 0 < norm m := zero_lt_one.trans_le hnm1 |
| have hmkN : norm m ≤ norm k := norm_le_of_dvd (primary_ne_zero hk.1) hmk |
| have hN0 : 0 < N := zero_lt_one.trans_le hN |
| have hM0 : 0 < M := hN0.trans_le hNM |
| have hM1 : 1 ≤ M := hN.trans hNM |
| have hMN1 : 1 ≤ M*N := one_le_mul_of_one_le_of_one_le hM1 hN |
| have hpow1 : 1 ≤ (M*N)^δ := Real.one_le_rpow hMN1 hδ.le |
| have hshape := sieveRecurrenceShape_nonneg (ξ := ξ) hM0.le hN0.le |
| have hMshape : M ≤ sieveRecurrenceShape ξ M N := by |
| unfold sieveRecurrenceShape |
| have hh : 0 ≤ (M*N)^(2/3 : ℝ) := by positivity |
| have hh' : 0 ≤ M^(1-ξ)*N^(2*ξ-1) := by positivity |
| linarith |
| change commonBlockCost S cubicSieveCutoff M N 8 k m ≤ _ |
| unfold commonBlockCost |
| split_ifs with hkN |
| · let n : ℝ := N/norm k |
| let z : ℝ := M/norm m |
| have hn : 1 ≤ n := by dsimp [n]; exact (one_le_div hnk).mpr hkN.le |
| have hn0 : 0 < n := zero_lt_one.trans_le hn |
| have hnz : n ≤ z := by |
| dsimp [n,z] |
| apply (div_le_div_iff₀ hnk hnm).mpr |
| nlinarith [mul_le_mul hNM hmkN (norm_nonneg m) hM0.le] |
| have hz : 0 < z := hn0.trans_le hnz |
| have hnN : n ≤ N := div_le_self hN0.le hnk1 |
| have hzM : z ≤ M := div_le_self hM0.le hnm1 |
| have hzn : z*n ≤ M*N := mul_le_mul hzM hnN hn0.le hM0.le |
| have hres : ∀ a ∈ residualRows S k, primary a ∧ Squarefree a ∧ norm a ≤ 2*n := by |
| intro a ha |
| have hh := residualRows_primary hk.1 hS' a ha |
| refine ⟨hh.1,hh.2,?_⟩ |
| have hnorm := (residualRows_norm hk.1 (fun p hp => (hS p hp).2.2) a ha).2 |
| have hsqrt : Real.sqrt 2 ≤ 2 := by norm_num |
| exact hnorm.trans (mul_le_mul_of_nonneg_right hsqrt hn0.le) |
| have hbound := hdual (residualRows S k) (2*n) (z/(27*n^2)) (by linarith) (by positivity) hres |
| have hnum := poisson_dual_numerical hξ hξ2 hδ hδ1 hn hnz |
| have herr : z^(1-ξ)*n^(2*ξ-1) ≤ M^(1-ξ)*N^(2*ξ-1) := |
| scaled_poisson_error_le hξ hM0 hN0 hnm1 hmkN |
| have hbracket : (z*n)^(2/3 : ℝ)+n+z^(1-ξ)*n^(2*ξ-1) ≤ sieveRecurrenceShape ξ M N := by |
| have hh : (z*n)^(2/3 : ℝ) ≤ (M*N)^(2/3 : ℝ) := Real.rpow_le_rpow (by positivity) hzn (by norm_num) |
| unfold sieveRecurrenceShape |
| linarith |
| calc |
| _ ≤ (z/n)*(D*(2*n)^δ* |
| ((2*n)*(z/(27*n^2))^(-(δ+1/3 : ℝ))+(2*n)^(2/3 : ℝ)*(z/(27*n^2))^(-(δ+2/3 : ℝ))+ |
| (z/(27*n^2))^(-(δ+ξ))+(z/(27*n^2))^(-(δ+1)))) := |
| mul_le_mul_of_nonneg_left hbound (by positivity) |
| _ = D*((z/n)*(2*n)^δ* |
| ((2*n)*(z/(27*n^2))^(-(δ+1/3 : ℝ))+(2*n)^(2/3 : ℝ)*(z/(27*n^2))^(-(δ+2/3 : ℝ))+ |
| (z/(27*n^2))^(-(δ+ξ))+(z/(27*n^2))^(-(δ+1)))) := by ring |
| _ ≤ D*(54000*(z*n)^δ*((z*n)^(2/3 : ℝ)+n+z^(1-ξ)*n^(2*ξ-1))) := |
| mul_le_mul_of_nonneg_left hnum hD.le |
| _ = (54000*D)*(z*n)^δ*((z*n)^(2/3 : ℝ)+n+z^(1-ξ)*n^(2*ξ-1)) := by ring |
| _ ≤ _ := by gcongr |
| · calc |
| _ ≤ E*(M/norm m) := hdiag _ (div_pos hM0 hnm) |
| _ ≤ C*M := mul_le_mul hEC (div_le_self hM0.le hnm1) (by positivity) hC.le |
| _ ≤ C*((M*N)^δ*sieveRecurrenceShape ξ M N) := mul_le_mul_of_nonneg_left |
| (hMshape.trans (le_mul_of_one_le_left hshape hpow1)) hC.le |
| _ = _ := by ring |
|
|
| lemma SieveExponent.thin_recurrence {ξ : ℝ} (hξ : 1 ≤ ξ) (hξ2 : ξ ≤ 2) |
| (hExp : SieveExponent ξ) {δ : ℝ} (hδ : 0 < δ) (hδ1 : δ ≤ 1) : |
| ∃ C : ℝ, 0 < C ∧ ∀ (S H : Finset Eisenstein) (M N : ℝ), 1 ≤ N → N ≤ M → |
| (∀ a ∈ S, primary a ∧ Squarefree a ∧ N ≤ norm a ∧ norm a ≤ Real.sqrt 2*N) → |
| (∀ x ∈ H, primary x ∧ norm x ≤ M) → |
| finiteCubicBound S H ≤ C*(M*N)^(2*δ)*sieveRecurrenceShape ξ M N := by |
| obtain ⟨C₀,hC₀,hrec⟩ := primarySmoothedSieveMass_norm_recurrence cubicSieveCutoff |
| cubicSieveCutoff_compact cubicSieveCutoff_smooth 8 (by norm_num) |
| obtain ⟨D,hD,hcost⟩ := hExp.common_cost_bound hξ hξ2 hδ hδ1 |
| obtain ⟨E,hE,henergy⟩ := common_energy_small_power δ hδ |
| refine ⟨C₀*D*E*(Real.sqrt 2)^δ,by positivity,?_⟩ |
| intro S H M N hN hNM hS hH |
| have hN0 : 0 < N := zero_lt_one.trans_le hN |
| have hM0 : 0 < M := hN0.trans_le hNM |
| have hM1 : 1 ≤ M := hN.trans hNM |
| have hS' : ∀ a ∈ S, primary a ∧ Squarefree a := fun a ha => ⟨(hS a ha).1,(hS a ha).2.1⟩ |
| have hshape := sieveRecurrenceShape_nonneg (ξ := ξ) hM0.le hN0.le |
| have hpow : (M*N)^δ*N^δ ≤ (M*N)^(2*δ) := by |
| rw [show 2*δ = δ+δ by ring,Real.rpow_add (by positivity)] |
| gcongr |
| exact le_mul_of_one_le_left hN0.le hM1 |
| apply (finiteCubicBound_le_iff _ _ (by positivity)).mpr |
| intro u |
| have hb := hrec S hS' u M N hM0 hN0 (fun a ha => (hS a ha).2.2) |
| have hsum : (∑ k ∈ commonRowFactors S, |
| (∑ s ∈ (primaryPrimeFactors k).powerset, |
| commonBlockCost S cubicSieveCutoff M N 8 k (∏ p ∈ s, p))*commonBlockEnergy S u k) ≤ |
| (D*(M*N)^δ*sieveRecurrenceShape ξ M N)* |
| ∑ k ∈ commonRowFactors S, (2 : ℝ)^(primaryPrimeFactors k).card*commonBlockEnergy S u k := by |
| rw [Finset.mul_sum] |
| apply Finset.sum_le_sum; intro k hk |
| have hh := Finset.sum_le_card_nsmul _ _ _ (hcost S M N hN hNM hS k hk) |
| simp only [Finset.card_powerset,nsmul_eq_mul,Nat.cast_pow,Nat.cast_ofNat] at hh |
| exact (mul_le_mul_of_nonneg_right hh (commonBlockEnergy_nonneg _ _ _)).trans_eq (by ring) |
| have he := henergy S u (Real.sqrt 2*N) |
| (fun a ha => ⟨(hS a ha).1,(hS a ha).2.1,(hS a ha).2.2.2⟩) |
| calc |
| _ ≤ ‖primarySmoothedSieveMass S u cubicSieveCutoff M‖ := |
| finiteCubicMass_le_smoothed S H (fun x hx => (hH x hx).1) u hM0 (fun x hx => (hH x hx).2) |
| _ ≤ C₀*((D*(M*N)^δ*sieveRecurrenceShape ξ M N)*(E*(Real.sqrt 2*N)^δ*∑ a ∈ S, ‖u a‖^2)) := |
| hb.trans (mul_le_mul_of_nonneg_left (hsum.trans (mul_le_mul_of_nonneg_left he (by positivity))) hC₀.le) |
| _ = (C₀*D*E*(Real.sqrt 2)^δ)*((M*N)^δ*N^δ)*sieveRecurrenceShape ξ M N*∑ a ∈ S, ‖u a‖^2 := by |
| rw [Real.mul_rpow (Real.sqrt_nonneg _) hN0.le]; ring |
| _ ≤ _ := by gcongr |
|
|
| end CubicFirstMoment |
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable |
| namespace CubicFirstMoment |
|
|
| lemma sieveRecurrenceShape_mono_right {ξ M n N : ℝ} (hξ : 1 ≤ ξ) |
| (hM : 0 ≤ M) (hn : 0 ≤ n) (hnN : n ≤ N) : |
| sieveRecurrenceShape ξ M n ≤ sieveRecurrenceShape ξ M N := by |
| unfold sieveRecurrenceShape |
| gcongr |
| linarith |
|
|
| lemma SieveExponent.recurrence {ξ : ℝ} (hξ : 1 ≤ ξ) (hξ2 : ξ ≤ 2) |
| (hExp : SieveExponent ξ) {ε : ℝ} (hε : 0 < ε) : |
| ∃ C : ℝ, 0 < C ∧ ∀ (S H : Finset Eisenstein) (M N : ℝ), 1 ≤ N → N ≤ M → |
| (∀ a ∈ S, primary a ∧ Squarefree a ∧ norm a ≤ N) → |
| (∀ x ∈ H, primary x ∧ norm x ≤ M) → |
| finiteCubicBound S H ≤ C*(M*N)^ε*sieveRecurrenceShape ξ M N := by |
| let δ : ℝ := min (ε/3) (1/3) |
| have hδ : 0 < δ := lt_min (by positivity) (by norm_num) |
| have hδ1 : δ ≤ 1 := (min_le_right _ _).trans (by norm_num) |
| have hδε : 3*δ ≤ ε := by have := min_le_left (ε/3) (1/3 : ℝ); change δ ≤ ε/3 at this; linarith |
| obtain ⟨D,hD,hthin⟩ := hExp.thin_recurrence hξ hξ2 hδ hδ1 |
| obtain ⟨E,hE,hlog⟩ := cutoff_log_small_power hδ |
| refine ⟨2*E*D,by positivity,?_⟩ |
| intro S H M N hN hNM hS hH |
| have hN0 : 0 < N := zero_lt_one.trans_le hN |
| have hM0 : 0 < M := hN0.trans_le hNM |
| have hM1 : 1 ≤ M := hN.trans hNM |
| have hshape := sieveRecurrenceShape_nonneg (ξ := ξ) hM0.le hN0.le |
| have hlocal : ∀ R : Finset Eisenstein, R ⊆ S → ∀ n : ℝ, 1 ≤ n → n ≤ N → |
| (∀ a ∈ R, n ≤ norm a ∧ norm a ≤ Real.sqrt 2*n) → |
| finiteCubicBound R H ≤ D*(M*N)^(2*δ)*sieveRecurrenceShape ξ M N := by |
| intro R hRS n hn hnN hR |
| have hh := hthin R H M n hn (hnN.trans hNM) |
| (fun a ha => ⟨(hS a (hRS ha)).1,(hS a (hRS ha)).2.1,hR a ha⟩) hH |
| have hs := sieveRecurrenceShape_mono_right hξ hM0.le (zero_lt_one.trans_le hn).le hnN |
| exact hh.trans (mul_le_mul (by gcongr) hs (sieveRecurrenceShape_nonneg hM0.le (by linarith)) (by positivity)) |
| have hb := finiteCubicBound_local_to_global S H N |
| (D*(M*N)^(2*δ)*sieveRecurrenceShape ξ M N) hN (by positivity) |
| (fun a ha => ⟨(hS a ha).1,(hS a ha).2.2⟩) (fun x hx => (hH x hx).1) hlocal |
| have hp : N^δ*(M*N)^(2*δ) ≤ (M*N)^ε := by |
| calc |
| _ ≤ (M*N)^δ*(M*N)^(2*δ) := by gcongr; exact le_mul_of_one_le_left hN0.le hM1 |
| _ = (M*N)^(3*δ) := by rw [← Real.rpow_add (by positivity)]; congr 1; ring |
| _ ≤ _ := Real.rpow_le_rpow_of_exponent_le (one_le_mul_of_one_le_of_one_le hM1 hN) hδε |
| calc |
| _ ≤ 2*((Nat.log 2 ⌊N⌋₊ : ℝ)+1)*(D*(M*N)^(2*δ)*sieveRecurrenceShape ξ M N) := hb |
| _ ≤ 2*(E*N^δ)*(D*(M*N)^(2*δ)*sieveRecurrenceShape ξ M N) := by gcongr; exact hlog N hN |
| _ = (2*E*D)*(N^δ*(M*N)^(2*δ))*sieveRecurrenceShape ξ M N := by ring |
| _ ≤ _ := by gcongr |
|
|
| lemma SieveExponent.mono {ξ ζ : ℝ} (hξ : SieveExponent ξ) (h : ξ ≤ ζ) : SieveExponent ζ := by |
| intro ε hε |
| obtain ⟨C,hC,hbound⟩ := hξ ε hε |
| refine ⟨C,hC,?_⟩ |
| intro S H M N hM hN hS hH |
| exact (hbound S H M N hM hN hS hH).trans (by |
| gcongr) |
|
|
| end CubicFirstMoment |
| end |
| end |
|
|
| end OAI |
|
|
