| import OAI.NumberTheory.PrimeGaps.SingularTails | |
| namespace OAI | |
| namespace LargePrimeGaps | |
| open Filter | |
| open Set Filter MeasureTheory | |
| open scoped Topology ContDiff | |
| noncomputable def distinctBox {s h : ℕ} (a : Fin s → ℤ) : Finset (Fin s → Fin h) := | |
| Finset.univ.filter fun n => Function.Injective (boxTuple a n) | |
| noncomputable def truncatedBoxMean {s h : ℕ} (a : Fin s → ℤ) (q : ℕ) : ℝ := | |
| (∑ n : Fin s → Fin h, siftedTuple (boxTuple a n) q)/(h:ℝ)^s | |
| noncomputable def distinctTruncatedBoxMean {s h : ℕ} (a : Fin s → ℤ) (q : ℕ) : ℝ := | |
| (∑ n ∈ distinctBox (h:=h) a, siftedTuple (boxTuple a n) q)/(h:ℝ)^s | |
| noncomputable def singularBoxMean {s h : ℕ} (a : Fin s → ℤ) : ℝ := | |
| (∑ n ∈ distinctBox (h:=h) a, singularSeries (Finset.univ.image (boxTuple a n)))/(h:ℝ)^s | |
| theorem siftedTuple_boxTuple {s h q : ℕ} [NeZero q] (a : Fin s → ℤ) (n : Fin s → Fin h) : | |
| siftedTuple (boxTuple a n) q = normalizedSiftedCount (residueTuple (q:=q) a n) := by | |
| rw [siftedTuple_eq] | |
| congr 1 | |
| funext i | |
| simp only [boxTuple, residueTuple, Int.cast_add, Int.cast_natCast] | |
| theorem truncatedBoxMean_bounds {s h q : ℕ} [NeZero q] (hh : 0<h) (hq : q≤h) | |
| (a : Fin s → ℤ) : | |
| (1-(q:ℝ)/h)^s ≤ truncatedBoxMean (h:=h) a q ∧ | |
| truncatedBoxMean (h:=h) a q ≤ (1+(q:ℝ)/h)^s := by | |
| have hhR : (0:ℝ)<h := by exact_mod_cast hh | |
| have hqR : (q:ℝ)≤h := by exact_mod_cast hq | |
| have hdiv : ((h/q:ℕ):ℝ)*(q:ℝ) ≤ h := by exact_mod_cast Nat.div_mul_le_self h q | |
| have hfloor : (h:ℝ)-(q:ℝ) ≤ ((h/q:ℕ):ℝ)*(q:ℝ) := by | |
| have he := Nat.div_add_mod h q | |
| have hm := Nat.mod_lt h (NeZero.pos q) | |
| have heR : ((h/q:ℕ):ℝ)*(q:ℝ)+(h%q:ℕ)=h := by | |
| exact_mod_cast (by simpa only [Nat.mul_comm] using he : h/q*q+h%q=h) | |
| have hmR : ((h%q:ℕ):ℝ)<q := by exact_mod_cast hm | |
| linarith | |
| have hb := residueTuple_sum_bounds (h:=h) a (@normalizedSiftedCount s q _) normalizedSiftedCount_nonneg | |
| rw [normalizedSiftedCount_sum] at hb | |
| simp_rw [← siftedTuple_boxTuple] at hb | |
| unfold truncatedBoxMean | |
| constructor | |
| · calc | |
| _ = ((h:ℝ)-(q:ℝ))^s/(h:ℝ)^s := by rw [← div_pow]; congr 1; field_simp | |
| _ ≤ (((h/q:ℕ):ℝ)*(q:ℝ))^s/(h:ℝ)^s := div_le_div_of_nonneg_right | |
| (pow_le_pow_left₀ (sub_nonneg.mpr hqR) hfloor s) (by positivity) | |
| _ ≤ _ := by rw [mul_pow]; exact div_le_div_of_nonneg_right hb.1 (by positivity) | |
| · calc | |
| _ ≤ (((h/q+1:ℕ):ℝ)*(q:ℝ))^s/(h:ℝ)^s := by | |
| rw [mul_pow]; exact div_le_div_of_nonneg_right hb.2 (by positivity) | |
| _ ≤ ((h:ℝ)+(q:ℝ))^s/(h:ℝ)^s := div_le_div_of_nonneg_right | |
| (pow_le_pow_left₀ (by positivity) (by push_cast; nlinarith) s) (by positivity) | |
| _ = _ := by rw [← div_pow]; congr 1; field_simp | |
| theorem distinctTruncatedBoxMean_nonneg {s h q : ℕ} [NeZero q] (a : Fin s → ℤ) : | |
| 0 ≤ distinctTruncatedBoxMean (h:=h) a q := by | |
| apply div_nonneg _ (by positivity) | |
| apply Finset.sum_nonneg | |
| intro n _ | |
| rw [siftedTuple_eq] | |
| exact normalizedSiftedCount_nonneg _ | |
| theorem distinctTruncatedBoxMean_le {s h q : ℕ} [NeZero q] (a : Fin s → ℤ) : | |
| distinctTruncatedBoxMean (h:=h) a q ≤ truncatedBoxMean (h:=h) a q := by | |
| apply div_le_div_of_nonneg_right _ (by positivity) | |
| apply Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _) | |
| intro n _ _ | |
| rw [siftedTuple_eq] | |
| exact normalizedSiftedCount_nonneg _ | |
| theorem truncatedBoxMean_sub_distinct_le {s h q : ℕ} [NeZero q] (hh : 0<h) | |
| (a : Fin s → ℤ) : | |
| truncatedBoxMean (h:=h) a q - distinctTruncatedBoxMean (h:=h) a q ≤ | |
| (s:ℝ)^2*(q:ℝ)^s/(h:ℝ) := by | |
| classical | |
| by_cases hs : s=0 | |
| · subst s | |
| have hall : distinctBox (h:=h) a = Finset.univ := by | |
| ext n | |
| simp [distinctBox, Function.Injective] | |
| simp [truncatedBoxMean, distinctTruncatedBoxMean, hall] | |
| have hhR : (0:ℝ)<h := by exact_mod_cast hh | |
| have hsum := Finset.sum_filter_add_sum_filter_not (Finset.univ : Finset (Fin s → Fin h)) | |
| (fun n => Function.Injective (boxTuple a n)) (fun n => siftedTuple (boxTuple a n) q) | |
| have hb : (∑ n ∈ Finset.univ.filter (fun n : Fin s → Fin h => ¬Function.Injective (boxTuple a n)), | |
| siftedTuple (boxTuple a n) q) ≤ (s:ℝ)^2*(h:ℝ)^(s-1)*(q:ℝ)^s := by | |
| calc | |
| _ ≤ ((Finset.univ.filter fun n : Fin s → Fin h => ¬Function.Injective (boxTuple a n)).card:ℝ) * | |
| (q:ℝ)^s := by | |
| simpa only [nsmul_eq_mul] using Finset.sum_le_card_nsmul _ _ ((q:ℝ)^s) (fun n _ => by | |
| rw [siftedTuple_eq] | |
| exact normalizedSiftedCount_le _) | |
| _ ≤ _ := mul_le_mul_of_nonneg_right (by exact_mod_cast boxTuple_nondistinct_count (h:=h) a) | |
| (by positivity) | |
| unfold truncatedBoxMean distinctTruncatedBoxMean distinctBox | |
| rw [← sub_div] | |
| calc | |
| _ ≤ ((s:ℝ)^2*(h:ℝ)^(s-1)*(q:ℝ)^s)/(h:ℝ)^s := by | |
| apply div_le_div_of_nonneg_right _ (by positivity) | |
| linarith [hsum] | |
| _ = _ := by | |
| have hp : (h:ℝ)^s=(h:ℝ)^(s-1)*(h:ℝ) := by | |
| rw [← pow_succ]; congr 1; omega | |
| rw [hp] | |
| field_simp | |
| noncomputable def boxTruncationError (s h q : ℕ) : ℝ := | |
| |(1-(q:ℝ)/h)^s-1| + |(1+(q:ℝ)/h)^s-1| + (s:ℝ)^2*(q:ℝ)^s/h | |
| theorem distinctTruncatedBoxMean_error {s h q : ℕ} [NeZero q] (hh : 0<h) (hq : q≤h) | |
| (a : Fin s → ℤ) : | |
| |distinctTruncatedBoxMean (h:=h) a q-1| ≤ boxTruncationError s h q := by | |
| have hb := truncatedBoxMean_bounds hh hq a | |
| have hd := truncatedBoxMean_sub_distinct_le (q:=q) hh a | |
| have hle := distinctTruncatedBoxMean_le (q:=q) (h:=h) a | |
| unfold boxTruncationError | |
| apply abs_le.mpr | |
| constructor | |
| · have hl := neg_abs_le ((1-(q:ℝ)/h)^s-1) | |
| have hpos := abs_nonneg ((1+(q:ℝ)/h)^s-1) | |
| linarith | |
| · have hu := le_abs_self ((1+(q:ℝ)/h)^s-1) | |
| have hpos := abs_nonneg ((1-(q:ℝ)/h)^s-1) | |
| have hnn : 0≤(s:ℝ)^2*(q:ℝ)^s/h := by positivity | |
| linarith | |
| noncomputable def uniformSingularTailBound (s M y : ℕ) : ℝ := | |
| 2*((s:ℝ)^2*(2:ℝ)^s)/(y:ℝ) + | |
| 4*(s:ℝ)*((s*(s-1):ℕ):ℝ)*Real.log M/((y:ℝ)*Real.log y) | |
| theorem singularTailBound_le_uniform {s M y : ℕ} (a : Fin s → ℤ) | |
| (ha : Function.Injective a) (hy : 1<y) | |
| (hd : ∀ i j, (a i-a j).natAbs ≤ M) : | |
| singularTailBound (Finset.univ.image a) y ≤ uniformSingularTailBound s M y := by | |
| have hc : (Finset.univ.image a).card=s := by | |
| rw [Finset.card_image_of_injective _ ha, Finset.card_univ, Fintype.card_fin] | |
| have hb := differenceProduct_le (Finset.univ.image a) (M:=M) (by | |
| intro x hx z hz | |
| obtain ⟨i, _, rfl⟩ := Finset.mem_image.mp hx | |
| obtain ⟨j, _, rfl⟩ := Finset.mem_image.mp hz | |
| exact hd i j) | |
| rw [hc] at hb | |
| have hl : Real.log (differenceProduct (Finset.univ.image a)) ≤ | |
| ((s*(s-1):ℕ):ℝ)*Real.log M := by | |
| rw [← Real.log_pow] | |
| apply Real.log_le_log | |
| · exact_mod_cast differenceProduct_pos (Finset.univ.image a) | |
| · exact_mod_cast hb | |
| unfold singularTailBound uniformSingularTailBound | |
| rw [hc] | |
| apply add_le_add le_rfl | |
| calc | |
| _ ≤ 4*(s:ℝ)*(((s*(s-1):ℕ):ℝ)*Real.log M)/((y:ℝ)*Real.log y) := | |
| div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_left hl (by positivity)) | |
| (mul_nonneg (by positivity) (Real.log_nonneg (by exact_mod_cast (show 1≤y by omega)))) | |
| _ = _ := by ring | |
| def boxDiameterBound {s h : ℕ} (a : Fin s → ℤ) (M : ℕ) : Prop := | |
| ∀ (n : Fin s → Fin h) i j, (boxTuple a n i-boxTuple a n j).natAbs ≤ M | |
| theorem uniformSingularTailBound_nonneg (s : ℕ) {M y : ℕ} (hM : 0<M) (hy : 1<y) : | |
| 0 ≤ uniformSingularTailBound s M y := by | |
| have hlogM : 0≤Real.log (M:ℝ) := Real.log_nonneg (by exact_mod_cast hM) | |
| have hlogy : 0≤Real.log (y:ℝ) := Real.log_nonneg (by exact_mod_cast (show 1≤y by omega)) | |
| unfold uniformSingularTailBound | |
| positivity | |
| theorem singularBoxMean_sub_truncation {s h M y : ℕ} (a : Fin s → ℤ) | |
| (hy : 1<y) (hys : 2*s≤y) (hd : boxDiameterBound (h:=h) a M) | |
| (hB : uniformSingularTailBound s M y≤1) : | |
| |singularBoxMean (h:=h) a-distinctTruncatedBoxMean (h:=h) a (primorial y)| ≤ | |
| 2*uniformSingularTailBound s M y*distinctTruncatedBoxMean (h:=h) a (primorial y) := by | |
| classical | |
| let : NeZero (primorial y) := ⟨(primorial_pos y).ne'⟩ | |
| have hpoint (n : Fin s → Fin h) (hn : n∈distinctBox a) : | |
| |singularSeries (Finset.univ.image (boxTuple a n))-siftedTuple (boxTuple a n) (primorial y)| ≤ | |
| 2*uniformSingularTailBound s M y*siftedTuple (boxTuple a n) (primorial y) := by | |
| have hi := (Finset.mem_filter.mp hn).2 | |
| have hc : (Finset.univ.image (boxTuple a n)).card=s := by | |
| rw [Finset.card_image_of_injective _ hi, Finset.card_univ, Fintype.card_fin] | |
| have hut := singularTailBound_le_uniform (boxTuple a n) hi hy (hd n) | |
| have ht := singularTail_error (Finset.univ.image (boxTuple a n)) hy (by simpa only [hc] using hys) | |
| (hut.trans hB) | |
| have hp : 0 ≤ siftedTuple (boxTuple a n) (primorial y) := by | |
| rw [siftedTuple_eq] | |
| exact normalizedSiftedCount_nonneg _ | |
| rw [singularSeries_eq_truncation_mul_tail _ hy (by simpa only [hc] using hys), | |
| ← siftedTuple_primorial_eq _ hi, ← primorial_eq_prod_primesLE] | |
| calc | |
| _ = siftedTuple (boxTuple a n) (primorial y)*|singularTail (Finset.univ.image (boxTuple a n)) y-1| := by | |
| rw [← mul_sub_one, abs_mul, abs_of_nonneg hp] | |
| _ ≤ siftedTuple (boxTuple a n) (primorial y)*(2*uniformSingularTailBound s M y) := | |
| mul_le_mul_of_nonneg_left (ht.trans (mul_le_mul_of_nonneg_left hut (by norm_num))) hp | |
| _ = _ := by ring | |
| unfold singularBoxMean distinctTruncatedBoxMean | |
| rw [← sub_div, abs_div, abs_of_nonneg (by positivity : 0≤(h:ℝ)^s), ← Finset.sum_sub_distrib] | |
| calc | |
| _ ≤ (∑ n ∈ distinctBox a, |singularSeries (Finset.univ.image (boxTuple a n))- | |
| siftedTuple (boxTuple a n) (primorial y)|)/(h:ℝ)^s := | |
| div_le_div_of_nonneg_right (Finset.abs_sum_le_sum_abs _ _) (by positivity) | |
| _ ≤ (∑ n ∈ distinctBox a, 2*uniformSingularTailBound s M y* | |
| siftedTuple (boxTuple a n) (primorial y))/(h:ℝ)^s := | |
| div_le_div_of_nonneg_right (Finset.sum_le_sum hpoint) (by positivity) | |
| _ = _ := by rw [← Finset.mul_sum]; ring | |
| theorem singularBoxMean_error {s h M y : ℕ} (a : Fin s → ℤ) (hh : 0<h) | |
| (hM : 0<M) (hy : 1<y) (hys : 2*s≤y) (hd : boxDiameterBound (h:=h) a M) | |
| (hB : uniformSingularTailBound s M y≤1) (hq : primorial y≤h) : | |
| |singularBoxMean (h:=h) a-1| ≤ boxTruncationError s h (primorial y) + | |
| 2*uniformSingularTailBound s M y*(1+(primorial y:ℝ)/h)^s := by | |
| let : NeZero (primorial y) := ⟨(primorial_pos y).ne'⟩ | |
| have hb := uniformSingularTailBound_nonneg s hM hy | |
| calc | |
| _ ≤ |singularBoxMean (h:=h) a-distinctTruncatedBoxMean (h:=h) a (primorial y)| + | |
| |distinctTruncatedBoxMean (h:=h) a (primorial y)-1| := abs_sub_le _ _ _ | |
| _ ≤ 2*uniformSingularTailBound s M y*distinctTruncatedBoxMean (h:=h) a (primorial y) + | |
| boxTruncationError s h (primorial y) := add_le_add | |
| (singularBoxMean_sub_truncation a hy hys hd hB) (distinctTruncatedBoxMean_error hh hq a) | |
| _ ≤ 2*uniformSingularTailBound s M y*(1+(primorial y:ℝ)/h)^s + | |
| boxTruncationError s h (primorial y) := add_le_add_left | |
| (mul_le_mul_of_nonneg_left ((distinctTruncatedBoxMean_le (q:=primorial y) a).trans (truncatedBoxMean_bounds hh hq a).2) | |
| (show 0 ≤ 2*uniformSingularTailBound s M y from mul_nonneg (by norm_num) hb)) _ | |
| _ = _ := by ring | |
| noncomputable def singularCutoffScale (s : ℕ) : ℝ := 2*((s:ℝ)+1)*Real.log 4 | |
| noncomputable def singularCutoff (s h : ℕ) : ℕ := ⌊Real.log h/singularCutoffScale s⌋₊ | |
| theorem singularCutoffScale_pos (s : ℕ) : 0<singularCutoffScale s := by | |
| unfold singularCutoffScale | |
| positivity | |
| theorem singularCutoff_tendsto (s : ℕ) : Tendsto (singularCutoff s) atTop atTop := | |
| tendsto_nat_floor_atTop.comp (Tendsto.atTop_div_const (singularCutoffScale_pos s) | |
| (Real.tendsto_log_atTop.comp tendsto_natCast_atTop_atTop)) | |
| theorem singularCutoff_ratio_tendsto (s : ℕ) : | |
| Tendsto (fun h : ℕ => (singularCutoff s h:ℝ)/Real.log h) atTop (𝓝 (1/singularCutoffScale s)) := by | |
| have hh := (tendsto_nat_floor_mul_div_atTop (a:=1/singularCutoffScale s) | |
| (one_div_nonneg.mpr (singularCutoffScale_pos s).le)).comp | |
| (Real.tendsto_log_atTop.comp (tendsto_natCast_atTop_atTop (R:=ℝ))) | |
| simpa only [singularCutoff, one_div_mul_eq_div, Function.comp_def] using hh | |
| theorem log_div_singularCutoff_tendsto (s : ℕ) : | |
| Tendsto (fun h : ℕ => Real.log h/(singularCutoff s h:ℝ)) atTop (𝓝 (singularCutoffScale s)) := by | |
| simpa only [inv_div, one_div, inv_inv] using (singularCutoff_ratio_tendsto s).inv₀ | |
| (one_div_ne_zero (singularCutoffScale_pos s).ne') | |
| theorem primorial_singularCutoff_power_bound (s : ℕ) {h : ℕ} (hh : 0<h) : | |
| (primorial (singularCutoff s h):ℝ)^(s+1) ≤ (h:ℝ)^(1/2:ℝ) := by | |
| have hhR : (0:ℝ)<h := by exact_mod_cast hh | |
| have hlog : 0≤Real.log (h:ℝ) := Real.log_nonneg (by exact_mod_cast hh) | |
| have hfloor := Nat.floor_le (div_nonneg hlog (singularCutoffScale_pos s).le) | |
| change (singularCutoff s h:ℝ) ≤ Real.log h/singularCutoffScale s at hfloor | |
| have hb := (le_div_iff₀ (singularCutoffScale_pos s)).mp hfloor | |
| have hexp : ((singularCutoff s h:ℝ)*((s:ℝ)+1))*Real.log 4 ≤ Real.log h*(1/2) := by | |
| unfold singularCutoffScale at hb | |
| nlinarith | |
| calc | |
| _ ≤ ((4:ℝ)^(singularCutoff s h))^(s+1) := pow_le_pow_left₀ (by positivity) | |
| (by exact_mod_cast primorial_le_four_pow (singularCutoff s h)) _ | |
| _ = Real.exp (((singularCutoff s h:ℝ)*((s:ℝ)+1))*Real.log 4) := by | |
| rw [← pow_mul, ← Real.exp_log (by norm_num : (0:ℝ)<4), ← Real.exp_nat_mul] | |
| rw [Real.log_exp] | |
| push_cast | |
| congr 1 | |
| _ ≤ _ := by rw [Real.rpow_def_of_pos hhR]; exact Real.exp_le_exp.mpr hexp | |
| theorem primorial_singularCutoff_ratio_tendsto (s r : ℕ) (hr : r ≤ s+1) : | |
| Tendsto (fun h : ℕ => (primorial (singularCutoff s h):ℝ)^r/(h:ℝ)) atTop (𝓝 0) := by | |
| have ht : Tendsto (fun h : ℕ => (h:ℝ)^(-(1/2:ℝ))) atTop (𝓝 0) := | |
| (tendsto_rpow_neg_atTop (by norm_num : (0:ℝ)<1/2)).comp tendsto_natCast_atTop_atTop | |
| apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds ht | |
| · exact Filter.Eventually.of_forall fun h => by positivity | |
| · filter_upwards [eventually_ge_atTop 1] with h hh | |
| have hhR : (0:ℝ)<h := by exact_mod_cast hh | |
| calc | |
| _ ≤ (primorial (singularCutoff s h):ℝ)^(s+1)/(h:ℝ) := | |
| div_le_div_of_nonneg_right (by exact_mod_cast (Nat.pow_le_pow_right | |
| (primorial_pos (singularCutoff s h)) hr)) hhR.le | |
| _ ≤ (h:ℝ)^(1/2:ℝ)/(h:ℝ) := div_le_div_of_nonneg_right | |
| (primorial_singularCutoff_power_bound s (by omega)) hhR.le | |
| _ = _ := by | |
| have he := Real.rpow_sub hhR (1/2:ℝ) 1 | |
| norm_num only [Real.rpow_one, show (1/2:ℝ)-1=-(1/2) by norm_num] at he | |
| exact he.symm | |
| theorem primorial_singularCutoff_div_tendsto (s : ℕ) : | |
| Tendsto (fun h : ℕ => (primorial (singularCutoff s h):ℝ)/(h:ℝ)) atTop (𝓝 0) := by | |
| simpa only [pow_one] using primorial_singularCutoff_ratio_tendsto s 1 (by omega) | |
| theorem log_mul_div_singularCutoff_tendsto (s : ℕ) {K : ℕ} (hK : 0<K) : | |
| Tendsto (fun h : ℕ => Real.log (K*h:ℕ)/(singularCutoff s h:ℝ)) atTop (𝓝 (singularCutoffScale s)) := by | |
| have hy := tendsto_natCast_atTop_atTop (R:=ℝ) |>.comp (singularCutoff_tendsto s) | |
| have hz : Tendsto (fun h : ℕ => Real.log (K:ℝ)/(singularCutoff s h:ℝ)) atTop (𝓝 0) := by | |
| simpa only [mul_zero, div_eq_mul_inv, Pi.inv_apply, Function.comp_apply] using hy.inv_tendsto_atTop.const_mul (Real.log (K:ℝ)) | |
| have ht := hz.add (log_div_singularCutoff_tendsto s) | |
| rw [zero_add] at ht | |
| apply ht.congr' | |
| filter_upwards [eventually_ge_atTop 1] with h hh | |
| rw [Nat.cast_mul, Real.log_mul (by exact_mod_cast hK.ne') (by exact_mod_cast (show h≠0 by omega)), add_div] | |
| theorem uniformSingularTailBound_tendsto (s : ℕ) {K : ℕ} (hK : 0<K) : | |
| Tendsto (fun h : ℕ => uniformSingularTailBound s (K*h) (singularCutoff s h)) atTop (𝓝 0) := by | |
| have hy := tendsto_natCast_atTop_atTop (R:=ℝ) |>.comp (singularCutoff_tendsto s) | |
| have hl := Real.tendsto_log_atTop.comp hy | |
| have ht := (hy.inv_tendsto_atTop.const_mul (2*((s:ℝ)^2*(2:ℝ)^s))).add | |
| (((log_mul_div_singularCutoff_tendsto s hK).mul hl.inv_tendsto_atTop).const_mul | |
| (4*(s:ℝ)*((s*(s-1):ℕ):ℝ))) | |
| simp only [mul_zero, add_zero] at ht | |
| convert ht using 1 | |
| ext h | |
| unfold uniformSingularTailBound | |
| simp only [div_eq_mul_inv, mul_inv_rev, Pi.inv_apply, Function.comp_apply] | |
| ring | |
| theorem boxTruncationError_tendsto (s : ℕ) : | |
| Tendsto (fun h : ℕ => boxTruncationError s h (primorial (singularCutoff s h))) atTop (𝓝 0) := by | |
| have hq := primorial_singularCutoff_div_tendsto s | |
| have hl := ((tendsto_const_nhds (x:=(1:ℝ)).sub hq).pow s).sub (tendsto_const_nhds (x:=(1:ℝ))) | |
| have hu := ((tendsto_const_nhds (x:=(1:ℝ)).add hq).pow s).sub (tendsto_const_nhds (x:=(1:ℝ))) | |
| have hd := (primorial_singularCutoff_ratio_tendsto s s (by omega)).const_mul ((s:ℝ)^2) | |
| have ht := (hl.abs.add hu.abs).add hd | |
| simp only [sub_zero, one_pow, sub_self, add_zero, abs_zero, mul_zero] at ht | |
| convert ht using 1 | |
| ext h | |
| unfold boxTruncationError | |
| ring | |
| noncomputable def singularBoxError (s K h : ℕ) : ℝ := | |
| boxTruncationError s h (primorial (singularCutoff s h)) + | |
| 2*uniformSingularTailBound s (K*h) (singularCutoff s h)* | |
| (1+(primorial (singularCutoff s h):ℝ)/h)^s | |
| theorem singularBoxError_tendsto (s : ℕ) {K : ℕ} (hK : 0<K) : | |
| Tendsto (singularBoxError s K) atTop (𝓝 0) := by | |
| have ht := (boxTruncationError_tendsto s).add | |
| (((uniformSingularTailBound_tendsto s hK).const_mul 2).mul | |
| ((tendsto_const_nhds (x:=(1:ℝ))).add (primorial_singularCutoff_div_tendsto s) |>.pow s)) | |
| simp only [mul_zero, zero_mul, add_zero] at ht | |
| exact ht | |
| theorem singularBoxMean_error_eventually (s : ℕ) {K : ℕ} (hK : 0<K) : | |
| ∀ᶠ h : ℕ in atTop, ∀ a : Fin s → ℤ, boxDiameterBound (h:=h) a (K*h) → | |
| |singularBoxMean (h:=h) a-1| ≤ singularBoxError s K h := by | |
| have hB := (uniformSingularTailBound_tendsto s hK).eventually (gt_mem_nhds (by norm_num : (0:ℝ)<1)) | |
| have hq := (primorial_singularCutoff_div_tendsto s).eventually (gt_mem_nhds (by norm_num : (0:ℝ)<1)) | |
| filter_upwards [eventually_ge_atTop 1, (singularCutoff_tendsto s).eventually_ge_atTop 2, | |
| (singularCutoff_tendsto s).eventually_ge_atTop (2*s), hB, hq] with h hh hy hys hB hq | |
| intro a hd | |
| have hqh : primorial (singularCutoff s h) ≤ h := by | |
| have hhR : (0:ℝ)<h := by exact_mod_cast hh | |
| have ht := (div_lt_one hhR).mp hq | |
| exact_mod_cast ht.le | |
| exact singularBoxMean_error a (by omega) (Nat.mul_pos hK (by omega)) (by omega) hys hd hB.le hqh | |
| theorem singularBoxMean_uniform (s : ℕ) {K : ℕ} (hK : 0<K) {ε : ℝ} (hε : 0<ε) : | |
| ∃ h₀ : ℕ, ∀ h≥h₀, ∀ a : Fin s → ℤ, boxDiameterBound (h:=h) a (K*h) → | |
| |singularBoxMean (h:=h) a-1| < ε := by | |
| have he := (singularBoxError_tendsto s hK).eventually (gt_mem_nhds hε) | |
| have hb := (singularBoxMean_error_eventually s hK).and he | |
| obtain ⟨h₀, hh₀⟩ := Filter.eventually_atTop.mp hb | |
| exact ⟨h₀, fun h hh a hd => ((hh₀ h hh).1 a hd).trans_lt (hh₀ h hh).2⟩ | |
| end LargePrimeGaps | |
| end OAI | |