import Mathlib import OAI.NumberTheory.Jacobsthal.Estimates.HeightBlockCount import OAI.NumberTheory.Jacobsthal.Estimates.HypotheticalPairRows namespace OAI namespace Erdos970 open scoped _root_.Erdos970 namespace ErdosCommonParameterInterval noncomputable def commonLower (a R T : ℝ) : ℝ := max (a/R) (a/T) noncomputable def commonUpper (b R T : ℝ) : ℝ := min (b/R) (b/T) theorem div_between_endpoints (a : ℝ) {R p T : ℝ} (hR : 0 < R) (hRp : R ≤ p) (hpT : p ≤ T) : min (a/R) (a/T) ≤ a/p ∧ a/p ≤ max (a/R) (a/T) := by have hp : 0 < p := hR.trans_le hRp have hT : 0 < T := hp.trans_le hpT by_cases ha : 0 ≤ a · exact ⟨(min_le_right _ _).trans (div_le_div_of_nonneg_left ha hp hpT), (div_le_div_of_nonneg_left ha hR hRp).trans (le_max_left _ _)⟩ · have han : 0 ≤ -a := by linarith have h1 := div_le_div_of_nonneg_left han hp hpT have h2 := div_le_div_of_nonneg_left han hR hRp simp only [neg_div] at h1 h2 exact ⟨(min_le_left _ _).trans (by linarith), (show a/p ≤ a/T by linarith).trans (le_max_right _ _)⟩ theorem common_interval_subset (a b : ℝ) {R p T : ℝ} (hR : 0 < R) (hRp : R ≤ p) (hpT : p ≤ T) : Set.Ioc (commonLower a R T) (commonUpper b R T) ⊆ Set.Ioc (a/p) (b/p) := by intro m hm exact ⟨(div_between_endpoints a hR hRp hpT).2.trans_lt hm.1, hm.2.trans (div_between_endpoints b hR hRp hpT).1⟩ theorem commonLower_error (a : ℝ) {R T : ℝ} (hR : 0 < R) (hRT : R ≤ T) : commonLower a R T ≤ a/T+|a| *(1/R-1/T) := by have hT : 0 < T := hR.trans_le hRT have hd : 0 ≤ 1/R-1/T := sub_nonneg.mpr (one_div_le_one_div_of_le hR hRT) have he : a/R=a/T+a*(1/R-1/T) := by ring apply max_le · rw [he] exact add_le_add le_rfl (mul_le_mul_of_nonneg_right (le_abs_self a) hd) · exact le_add_of_nonneg_right (mul_nonneg (abs_nonneg a) hd) theorem commonUpper_error (b : ℝ) {R T : ℝ} (hR : 0 < R) (hRT : R ≤ T) : b/T-|b| *(1/R-1/T) ≤ commonUpper b R T := by have hT : 0 < T := hR.trans_le hRT have hd : 0 ≤ 1/R-1/T := sub_nonneg.mpr (one_div_le_one_div_of_le hR hRT) have he : b/R=b/T+b*(1/R-1/T) := by ring apply le_min · rw [he] have h := mul_le_mul_of_nonneg_right (neg_abs_le b) hd nlinarith · exact sub_le_self _ (mul_nonneg (abs_nonneg b) hd) theorem common_length_lower (a b : ℝ) {R T : ℝ} (hR : 0 < R) (hRT : R ≤ T) : (b-a)/T-(|a|+|b|)*(1/R-1/T) ≤ commonUpper b R T-commonLower a R T := by have hl := commonLower_error a hR hRT have hu := commonUpper_error b hR hRT have he : (b-a)/T-(|a|+|b|)*(1/R-1/T) = (b/T-|b| *(1/R-1/T))-(a/T+|a| *(1/R-1/T)) := by ring rw [he] exact sub_le_sub hu hl theorem common_height_bound (a b m : ℝ) {R T : ℝ} (hR : 0 < R) (hm : m ∈ Set.Ioc (commonLower a R T) (commonUpper b R T)) : |m| ≤ max |a| |b|/R := by apply abs_le.mpr constructor · have h1 : -(max |a| |b|) ≤ a := (neg_le_neg (le_max_left _ _)).trans (neg_abs_le a) have h2 := div_le_div_of_nonneg_right h1 hR.le have h3 : a/R ≤ commonLower a R T := le_max_left _ _ rw [neg_div] at h2 exact h2.trans (h3.trans hm.1.le) · have h1 : b ≤ max |a| |b| := (le_abs_self b).trans (le_max_right _ _) have h2 := div_le_div_of_nonneg_right h1 hR.le exact hm.2.trans ((min_le_left _ _).trans h2) theorem reciprocal_width_bound {R T epsilon : ℝ} (hR : 0 < R) (hRT : R ≤ T) (hwidth : T ≤ (1+epsilon)*R) : 1/R-1/T ≤ epsilon/T := by have hT : 0 < T := hR.trans_le hRT have hdiv : T/R ≤ 1+epsilon := (div_le_iff₀ hR).mpr hwidth have he : (1/R-1/T)*T=T/R-1 := by field_simp apply (le_div_iff₀ hT).mpr rw [he] linarith theorem common_length_lower_relative (a b : ℝ) {R T epsilon : ℝ} (hR : 0 < R) (hRT : R ≤ T) (hwidth : T ≤ (1+epsilon)*R) : ((b-a)-epsilon*(|a|+|b|))/T ≤ commonUpper b R T-commonLower a R T := by have hd := reciprocal_width_bound hR hRT hwidth have hmul := mul_le_mul_of_nonneg_left hd (add_nonneg (abs_nonneg a) (abs_nonneg b)) calc _ = (b-a)/T-(|a|+|b|)*(epsilon/T) := by ring _ ≤ (b-a)/T-(|a|+|b|)*(1/R-1/T) := sub_le_sub_left hmul _ _ ≤ _ := common_length_lower a b hR hRT theorem affine_common_interval (A D Y q m : ℝ) {R p T : ℝ} (hq : 0 < q) (hR : 0 < R) (hRp : R ≤ p) (hpT : p ≤ T) (hm : m ∈ Set.Ioc (commonLower (-A/q) R T) (commonUpper ((D*Y-A)/q) R T)) : 0 < A+p*q*m ∧ A+p*q*m ≤ D*Y := by have hp : 0 < p := hR.trans_le hRp have hc := common_interval_subset (-A/q) ((D*Y-A)/q) hR hRp hpT hm simp only [div_div] at hc have hl := (div_lt_iff₀ (mul_pos hq hp)).mp hc.1 have hu := (le_div_iff₀ (mul_pos hq hp)).mp hc.2 constructor <;> nlinarith end ErdosCommonParameterInterval namespace ErdosCommonMInterval open ErdosCommonParameterInterval noncomputable def lower (A q R T : ℝ) : ℝ := commonLower (-A/q) R T + 1 noncomputable def upper (A D Y q R T : ℝ) : ℝ := commonUpper ((D*Y-A)/q) R T noncomputable def interval (A D Y q R T : ℝ) : Set ℝ := Set.Icc (lower A q R T) (upper A D Y q R T) noncomputable def referenceLength (Y R q : ℝ) : ℝ := Y/(R*q) noncomputable def lengthExponent (P Y R q : ℝ) : ℝ := Real.log (referenceLength Y R q)/Real.log P theorem closed_subset (A D Y q R T : ℝ) : interval A D Y q R T ⊆ Set.Ioc (commonLower (-A/q) R T) (commonUpper ((D*Y-A)/q) R T) := by intro m hm exact ⟨by have := hm.1; dsimp [interval,lower] at this; linarith, hm.2⟩ theorem closed_affine {A D Y q R T m p : ℝ} (hq : 0 < q) (hR : 0 < R) (hRp : R ≤ p) (hpT : p ≤ T) (hm : m ∈ interval A D Y q R T) : 0 < A+p*q*m ∧ A+p*q*m ≤ D*Y := affine_common_interval A D Y q m hq hR hRp hpT (closed_subset A D Y q R T hm) theorem integer_decode {A m : ℤ} {D Y q p : ℕ} {R T : ℝ} (hD : 0 < D) (hq : 0 < q) (hR : 0 < R) (hRp : R ≤ (p : ℝ)) (hpT : (p : ℝ) ≤ T) (hm : (m : ℝ) ∈ interval A D Y q R T) (hdvd : (D : ℤ) ∣ A+(p : ℤ)*q*m) : ∃ n : ℕ, 1 ≤ n ∧ n ≤ Y ∧ (D : ℤ)*n=A+(p : ℤ)*q*m := by have hb := closed_affine (by exact_mod_cast hq) hR hRp hpT hm obtain ⟨k,hk⟩ := hdvd have he : (A : ℝ)+(p : ℝ)*q*m=(D : ℝ)*(k : ℝ) := by exact_mod_cast hk have hDp : (0 : ℝ) < D := by exact_mod_cast hD have hkpos : (0 : ℝ) < k := by nlinarith [hb.1] have hkY : (k : ℝ) ≤ Y := by nlinarith [hb.2] have hkint : 0 ≤ k := le_of_lt (by exact_mod_cast hkpos : (0 : ℤ) < k) refine ⟨k.toNat, ?_, ?_, ?_⟩ · have : (1 : ℤ) ≤ k := by exact_mod_cast (by exact_mod_cast hkpos : (0 : ℤ) < k) exact_mod_cast (show (1 : ℤ) ≤ (k.toNat : ℤ) by simpa [Int.toNat_of_nonneg hkint] using this) · have : k ≤ (Y : ℤ) := by exact_mod_cast hkY exact_mod_cast (show (k.toNat : ℤ) ≤ (Y : ℤ) by simpa [Int.toNat_of_nonneg hkint] using this) · simpa [Int.toNat_of_nonneg hkint] using hk.symm def decode (A m : ℤ) (D q p : ℕ) : ℕ := ((A+(p : ℤ)*q*m)/(D : ℤ)).toNat theorem decode_spec {A m : ℤ} {D Y q p : ℕ} {R T : ℝ} (hD : 0 < D) (hq : 0 < q) (hR : 0 < R) (hRp : R ≤ (p : ℝ)) (hpT : (p : ℝ) ≤ T) (hm : (m : ℝ) ∈ interval A D Y q R T) (hdvd : (D : ℤ) ∣ A+(p : ℤ)*q*m) : 1 ≤ decode A m D q p ∧ decode A m D q p ≤ Y ∧ (D : ℤ)*decode A m D q p=A+(p : ℤ)*q*m := by obtain ⟨n,hn1,hnY,hn⟩ := integer_decode hD hq hR hRp hpT hm hdvd have hDn : (D : ℤ) ≠ 0 := by exact_mod_cast hD.ne' have hid : decode A m D q p=n := by unfold decode rw [← hn,Int.mul_ediv_cancel_left _ hDn] simp simpa only [hid] using And.intro hn1 (And.intro hnY hn) theorem reference_rpow {P Y R q : ℝ} (hP : 1 < P) (hY : 0 < Y) (hR : 0 < R) (hq : 0 < q) : P ^ lengthExponent P Y R q = referenceLength Y R q := by have hJ : 0 < referenceLength Y R q := div_pos hY (mul_pos hR hq) rw [lengthExponent,Real.rpow_def_of_pos (by linarith : 0 < P)] rw [mul_div_cancel₀ _ (Real.log_pos hP).ne'] exact Real.exp_log hJ end ErdosCommonMInterval namespace ErdosCommonMInterval open ErdosCommonParameterInterval theorem endpoint_magnitude {A D Y q : ℝ} (hD : 0 ≤ D) (hY : 0 ≤ Y) (hq : 0 < q) : |-A/q|+|(D*Y-A)/q| ≤ (D*Y+2*|A|)/q := by rw [abs_div,abs_div,abs_of_pos hq,abs_neg,← add_div] apply div_le_div_of_nonneg_right _ hq.le have h : |D*Y-A| ≤ |D*Y|+|A| := by simpa only [sub_zero, zero_sub, abs_neg] using abs_sub_le (D*Y) 0 A rw [abs_of_nonneg (mul_nonneg hD hY)] at h linarith theorem closed_length_lower {A D Y q R T H e : ℝ} (hD : 1 ≤ D) (hY : 0 < Y) (hq : 0 < q) (hR : 0 < R) (hRT : R ≤ T) (hH : 0 ≤ H) (he : 0 ≤ e) (hwidth : T ≤ (1+e)*R) (hA : |A| ≤ Y*H) : (1-2*e*(1+H))*D*referenceLength Y R q-1 ≤ upper A D Y q R T-lower A q R T := by have hT : 0 < T := hR.trans_le hRT have hD0 : 0 < D := by linarith have hd0 : 0 ≤ 1/R-1/T := sub_nonneg.mpr (one_div_le_one_div_of_le hR hRT) have hd := (reciprocal_width_bound hR hRT hwidth).trans (div_le_div_of_nonneg_left he hR hRT) have hmag := endpoint_magnitude (A := A) hD0.le hY.le hq have hbase : (D*Y-A)/q-(-A/q) = D*Y/q := by ring have hnorm : D*Y/q+|-A/q|+|(D*Y-A)/q| ≤ 2*D*Y*(1+H)/q := by have hDH : Y*H ≤ D*Y*H := by nlinarith [mul_nonneg hY.le hH] have : (D*Y+2*|A|)/q ≤ (D*Y+2*D*Y*H)/q := div_le_div_of_nonneg_right (by linarith) hq.le calc _ ≤ D*Y/q+(D*Y+2*D*Y*H)/q := by linarith _ = _ := by ring have hnorm0 : 0 ≤ D*Y/q+|-A/q|+|(D*Y-A)/q| := by positivity have hprod := mul_le_mul hnorm hd hd0 (by positivity : 0 ≤ 2*D*Y*(1+H)/q) have hf := common_length_lower (-A/q) ((D*Y-A)/q) hR hRT rw [hbase] at hf have heq : (D*Y/q)/T-(|-A/q|+|(D*Y-A)/q|)*(1/R-1/T) = D*Y/(R*q)-(D*Y/q+|-A/q|+|(D*Y-A)/q|)*(1/R-1/T) := by ring rw [heq] at hf dsimp [upper,lower,referenceLength] have heq2 : D*Y/(R*q)-(2*D*Y*(1+H)/q)*(e/R) = (1-2*e*(1+H))*D*(Y/(R*q)) := by ring linarith theorem closed_length_upper (A D Y q R T : ℝ) : upper A D Y q R T-lower A q R T ≤ D*referenceLength Y R q-1 := by have hU : upper A D Y q R T ≤ ((D*Y-A)/q)/R := min_le_left _ _ have hL : (-A/q)/R+1 ≤ lower A q R T := by have h : (-A/q)/R ≤ ErdosCommonParameterInterval.commonLower (-A/q) R T := le_max_left _ _ dsimp [lower];linarith have heq : ((D*Y-A)/q)/R-((-A/q)/R+1)=D*referenceLength Y R q-1 := by dsimp [referenceLength];ring linarith theorem closed_height {A D Y q R T H m : ℝ} (hD : 0 ≤ D) (hY : 0 < Y) (hq : 0 < q) (hR : 0 < R) (hA : |A| ≤ Y*H) (hm : m ∈ interval A D Y q R T) : |m| ≤ (D+H)*referenceLength Y R q := by have hb := common_height_bound (-A/q) ((D*Y-A)/q) m hR (closed_subset A D Y q R T hm) have ha : |-A/q| ≤ (D*Y+Y*H)/q := by rw [abs_div,abs_neg,abs_of_pos hq] exact div_le_div_of_nonneg_right (by nlinarith) hq.le have hc : |(D*Y-A)/q| ≤ (D*Y+Y*H)/q := by rw [abs_div,abs_of_pos hq] apply div_le_div_of_nonneg_right _ hq.le have h : |D*Y-A| ≤ |D*Y|+|A| := by simpa only [sub_zero, zero_sub, abs_neg] using abs_sub_le (D*Y) 0 A rw [abs_of_nonneg (mul_nonneg hD hY.le)] at h linarith calc _ ≤ max |-A/q| |(D*Y-A)/q|/R := hb _ ≤ ((D*Y+Y*H)/q)/R := div_le_div_of_nonneg_right (max_le ha hc) hR.le _ = _ := by dsimp [referenceLength];ring end ErdosCommonMInterval section open _root_.Filter open scoped Topology namespace ErdosCommonMInterval theorem eventual_polynomial_cutoff (B K : ℝ) (hB : 0 ≤ B) (hK : 0 < K) : ∀ᶠ w : ℝ in atTop, 2 ≤ w ∧ w^B ≤ Real.exp (K*(Real.log w)^3) := by filter_upwards [eventually_ge_atTop (2 : ℝ), Real.tendsto_log_atTop.eventually_ge_atTop (max 1 (B/K))] with w hw hL have hw0 : 0 < w := by linarith have hL1 : 1 ≤ Real.log w := (le_max_left _ _).trans hL have hBL : B ≤ K*Real.log w := by have := (div_le_iff₀ hK).mp (show B/K ≤ Real.log w from (le_max_right _ _).trans hL) nlinarith refine ⟨hw, ?_⟩ rw [Real.rpow_def_of_pos hw0] apply Real.exp_le_exp.mpr have hsq : Real.log w ≤ (Real.log w)^2 := by nlinarith have hpow : (Real.log w)^2 ≤ (Real.log w)^3 := by nlinarith nlinarith [mul_le_mul_of_nonneg_right hBL (by linarith : 0 ≤ Real.log w)] theorem subbin_loss {w C : ℝ} (hw : 1 ≤ w) (hC : 0 ≤ C) : 2*w^(-2*C-10)*(1+w^(C+2)) ≤ 4*w^(-8 : ℝ) := by have hw0 : 0 < w := by linarith have hH : 1 ≤ w^(C+2) := Real.one_le_rpow hw (by linarith) have hmul : w^(-2*C-10)*w^(C+2) = w^(-C-8) := by rw [← Real.rpow_add hw0];congr 1;ring have hcmp : w^(-C-8) ≤ w^(-8 : ℝ) := Real.rpow_le_rpow_of_exponent_le hw (by linarith) have hnon : 0 ≤ w^(-2*C-10) := Real.rpow_nonneg hw0.le _ have hle : w^(-2*C-10) ≤ w^(-2*C-10)*w^(C+2) := by nlinarith nlinarith theorem height_absorption {w P C : ℝ} (hw : 2 ≤ w) (hP : w^(50*(C+3)) ≤ P) : 2*w^(C+2) ≤ P^((2 : ℝ)/100) := by have hw0 : 0 < w := by linarith have h1 : 2*w^(C+2) ≤ w^(C+3) := by have he : w^(C+3)=w*w^(C+2) := by calc w^(C+3) = w^((1 : ℝ)+(C+2)) := by congr 1;ring _ = w*w^(C+2) := by rw [Real.rpow_add hw0,Real.rpow_one] rw [he] exact mul_le_mul_of_nonneg_right hw (Real.rpow_nonneg hw0.le _) have h2 := Real.rpow_le_rpow (Real.rpow_nonneg hw0.le (50*(C+3))) hP (by norm_num : (0 : ℝ) ≤ 2/100) have he : (w^(50*(C+3)))^((2 : ℝ)/100)=w^(C+3) := by rw [← Real.rpow_mul hw0.le];congr 1;ring exact h1.trans (by rwa [he] at h2) noncomputable def relativeError (w : ℝ) : ℝ := 4*w^(-8 : ℝ)+w^(-2 : ℝ) theorem relativeError_tends_zero : Tendsto relativeError atTop (𝓝 0) := by have h8 := tendsto_rpow_neg_atTop (by norm_num : (0 : ℝ) < 8) have h2 := tendsto_rpow_neg_atTop (by norm_num : (0 : ℝ) < 2) change Tendsto (fun w : ℝ => 4*w^(-8 : ℝ)+w^(-2 : ℝ)) atTop (𝓝 0) simpa only [mul_zero,add_zero] using (h8.const_mul 4).add h2 theorem uniform_common_interval (C K delta : ℝ) (hC : 0 ≤ C) (hK : 0 < K) (hd0 : 0 < delta) (hd1 : delta < 1) : ∃ W : ℝ, ∀ w : ℝ, W ≤ w → ∀ P A D Y q R T : ℝ, Real.exp (K*(Real.log w)^3) ≤ P → 1 ≤ D → D ≤ w^C → 0 < Y → 0 < q → 0 < R → R ≤ T → T ≤ (1+w^(-2*C-10))*R → |A| ≤ Y*w^(C+2) → 2 ≤ lengthExponent P Y R q → (1-delta)*D*referenceLength Y R q ≤ upper A D Y q R T-lower A q R T ∧ upper A D Y q R T-lower A q R T ≤ D*referenceLength Y R q ∧ (interval A D Y q R T).Nonempty ∧ ∀ m ∈ interval A D Y q R T, |m| ≤ P^(lengthExponent P Y R q+(2 : ℝ)/100) := by have hE := relativeError_tends_zero.eventually (gt_mem_nhds hd0) have hpoly := eventual_polynomial_cutoff (50*(C+3)) K (by positivity) hK obtain ⟨W,hW⟩ := eventually_atTop.mp (hE.and hpoly) refine ⟨W, ?_⟩ intro w hw P A D Y q R T hP hD hDu hY hq hR hRT hwidth hA hs obtain ⟨hErr,hw2,hpow⟩ := hW w hw have hw0 : 0 < w := by linarith have hw1 : 1 ≤ w := by linarith have hwpow : w ≤ w^(50*(C+3)) := by calc w = w^(1 : ℝ) := (Real.rpow_one _).symm _ ≤ _ := Real.rpow_le_rpow_of_exponent_le hw1 (by nlinarith) have hwP : w ≤ P := hwpow.trans (hpow.trans hP) have hP1 : 1 < P := by linarith have hD0 : 0 < D := by linarith have hJ : 0 < referenceLength Y R q := div_pos hY (mul_pos hR hq) have hJid := reference_rpow hP1 hY hR hq have hJlow : P^2 ≤ referenceLength Y R q := by rw [← hJid] simpa only [Real.rpow_ofNat] using Real.rpow_le_rpow_of_exponent_le hP1.le hs have hJw : w^2 ≤ referenceLength Y R q := by nlinarith have hpay : 1 ≤ w^(-2 : ℝ)*D*referenceLength Y R q := by have hid : w^(-2 : ℝ)*w^2=1 := by rw [← Real.rpow_ofNat,← Real.rpow_add hw0];norm_num have hn : 0 ≤ w^(-2 : ℝ) := Real.rpow_nonneg hw0.le _ have hm : w^2 ≤ D*referenceLength Y R q := by nlinarith have := mul_le_mul_of_nonneg_left hm hn nlinarith have hraw := closed_length_lower hD hY hq hR hRT (Real.rpow_nonneg hw0.le (C+2)) (Real.rpow_nonneg hw0.le (-2*C-10)) hwidth hA have hcost := subbin_loss hw1 hC have hlen : (1-delta)*D*referenceLength Y R q ≤ upper A D Y q R T-lower A q R T := by have hmul := mul_le_mul_of_nonneg_right hcost (mul_pos hD0 hJ).le have herrmul := mul_le_mul_of_nonneg_right hErr.le (mul_pos hD0 hJ).le dsimp [relativeError] at herrmul nlinarith refine ⟨hlen, (closed_length_upper A D Y q R T).trans (by linarith), ?_, ?_⟩ · apply Set.nonempty_Icc.mpr have := mul_pos (mul_pos (by linarith : 0 < 1-delta) hD0) hJ linarith · intro m hm have hheight := closed_height hD0.le hY hq hR hA hm have hDC : D ≤ w^(C+2) := hDu.trans (Real.rpow_le_rpow_of_exponent_le hw1 (by linarith)) have habs := height_absorption hw2 (hpow.trans hP) have hp02 : D+w^(C+2) ≤ P^((2 : ℝ)/100) := by linarith calc |m| ≤ (D+w^(C+2))*referenceLength Y R q := hheight _ ≤ P^((2 : ℝ)/100)*referenceLength Y R q := mul_le_mul_of_nonneg_right hp02 hJ.le _ = P^(lengthExponent P Y R q+(2 : ℝ)/100) := by rw [← hJid,← Real.rpow_add (by linarith : 0 < P)];congr 1;ring theorem uniform_integer_common_interval (C K delta : ℝ) (hC : 0 ≤ C) (hK : 0 < K) (hd0 : 0 < delta) (hd1 : delta < 1) : ∃ W : ℝ, ∀ w : ℝ, W ≤ w → ∀ P R T : ℝ, ∀ A : ℤ, ∀ D Y q : ℕ, Real.exp (K*(Real.log w)^3) ≤ P → 0 < D → (D : ℝ) ≤ w^C → 0 < Y → 0 < q → 0 < R → R ≤ T → T ≤ (1+w^(-2*C-10))*R → |(A : ℝ)| ≤ (Y : ℝ)*w^(C+2) → 2 ≤ lengthExponent P Y R q → (1-delta)*(D : ℝ)*referenceLength Y R q ≤ upper A D Y q R T-lower A q R T ∧ upper A D Y q R T-lower A q R T ≤ (D : ℝ)*referenceLength Y R q ∧ (interval A D Y q R T).Nonempty ∧ (∀ m ∈ interval A D Y q R T, |m| ≤ P^(lengthExponent P Y R q+(2 : ℝ)/100)) ∧ ∀ p : ℕ, ∀ m : ℤ, R ≤ (p : ℝ) → (p : ℝ) ≤ T → (m : ℝ) ∈ interval A D Y q R T → (D : ℤ) ∣ A+(p : ℤ)*q*m → 1 ≤ decode A m D q p ∧ decode A m D q p ≤ Y ∧ (D : ℤ)*decode A m D q p=A+(p : ℤ)*q*m := by obtain ⟨W,hW⟩ := uniform_common_interval C K delta hC hK hd0 hd1 refine ⟨W, ?_⟩ intro w hw P R T A D Y q hP hD hDu hY hq hR hRT hwidth hA hs have hb := hW w hw P A D Y q R T hP (by exact_mod_cast (show 1 ≤ D from hD)) hDu (by exact_mod_cast hY) (by exact_mod_cast hq) hR hRT hwidth hA hs refine ⟨hb.1,hb.2.1,hb.2.2.1,hb.2.2.2,?_⟩ intro p m hRp hpT hm hdvd exact decode_spec hD hq hR hRp hpT hm hdvd end ErdosCommonMInterval end namespace ErdosLargeHeightBlocks noncomputable def blockLeft (C0 : ℤ) (R xi : ℝ) (H i : ℕ) : ℝ := ErdosTagSubbins.left R xi (blockCount C0 R xi H) i noncomputable def blockRight (C0 : ℤ) (R xi : ℝ) (H i : ℕ) : ℝ := ErdosTagSubbins.right R xi (blockCount C0 R xi H) i theorem block_reciprocal_variation (C0 : ℤ) (R xi : ℝ) (H i : ℕ) (hR : 0 < R) (hxi : 0 < xi) (hH : 0 < H) : |(C0 : ℝ)/blockLeft C0 R xi H i-(C0 : ℝ)/blockRight C0 R xi H i| ≤ (H : ℝ) := by let N := blockCount C0 R xi H let a := blockLeft C0 R xi H i let b := blockRight C0 R xi H i have hN : 0 < N := blockCount_pos C0 R xi H have hNr : (0 : ℝ) < N := by exact_mod_cast hN have hHr : (0 : ℝ) < H := by exact_mod_cast hH have hs := ErdosTagSubbins.step_pos hR hxi hN have ha : R ≤ a := ErdosTagSubbins.left_lower hs.le have hab : a < b := ErdosTagSubbins.left_lt_right hR hxi hN have hb : R ≤ b := ha.trans hab.le have ha0 : 0 < a := hR.trans_le ha have hb0 : 0 < b := hR.trans_le hb have hd : b-a = ErdosTagSubbins.step R xi N := ErdosTagSubbins.right_sub_left R xi N i have hden : R^2 ≤ a*b := by simpa only [pow_two] using mul_le_mul ha hb hR.le ha0.le have he : (C0 : ℝ)/a-(C0 : ℝ)/b=(C0 : ℝ)*(b-a)/(a*b) := by field_simp have hcount := (div_le_iff₀ (mul_pos hR hHr)).mp (blockCount_lower C0 R xi H) change |(C0 : ℝ)/a-(C0 : ℝ)/b| ≤ (H : ℝ) rw [he,abs_div,abs_mul,abs_of_nonneg (sub_nonneg.mpr hab.le),abs_of_pos (mul_pos ha0 hb0)] calc _ ≤ |(C0 : ℝ)| * (b-a)/R^2 := div_le_div_of_nonneg_left (mul_nonneg (abs_nonneg _) (sub_nonneg.mpr hab.le)) (sq_pos_of_pos hR) hden _ = xi*|(C0 : ℝ)|/(R*(N : ℝ)) := by rw [hd] dsimp [ErdosTagSubbins.step] field_simp _ ≤ _ := by apply (div_le_iff₀ (mul_pos hR hNr)).mpr change xi*|(C0 : ℝ)| ≤ (N : ℝ)*(R*(H : ℝ)) at hcount nlinarith noncomputable def coordinateLower (C0 : ℤ) (R xi : ℝ) (H i : ℕ) : ℝ := min ((C0 : ℝ)/blockLeft C0 R xi H i) ((C0 : ℝ)/blockRight C0 R xi H i) noncomputable def coordinateUpper (C0 : ℤ) (R xi : ℝ) (H i : ℕ) : ℝ := max ((C0 : ℝ)/blockLeft C0 R xi H i) ((C0 : ℝ)/blockRight C0 R xi H i)+(H : ℝ) theorem block_coordinate_containment (C0 : ℤ) (R xi : ℝ) (H i : ℕ) (hR : 0 < R) (hxi : 0 < xi) (p m : ℝ) (hpL : blockLeft C0 R xi H i ≤ p) (hpU : p ≤ blockRight C0 R xi H i) (hmL : (C0 : ℝ)/p ≤ m) (hmU : m < (C0 : ℝ)/p+(H : ℝ)) : m ∈ Set.Ico (coordinateLower C0 R xi H i) (coordinateUpper C0 R xi H i) := by have hL : 0 < blockLeft C0 R xi H i := ErdosTagSubbins.left_pos hR hxi (blockCount_pos C0 R xi H) have hb := ErdosCommonParameterInterval.div_between_endpoints (C0 : ℝ) hL hpL hpU refine ⟨hb.1.trans hmL,?_⟩ dsimp only [coordinateUpper] linarith [hb.2] theorem block_coordinate_width (C0 : ℤ) (R xi : ℝ) (H i : ℕ) (hR : 0 < R) (hxi : 0 < xi) (hH : 0 < H) : coordinateUpper C0 R xi H i-coordinateLower C0 R xi H i ≤ 2*(H : ℝ) := by have hv := block_reciprocal_variation C0 R xi H i hR hxi hH have he (x y : ℝ) : max x y-min x y=|x-y| := by rw [max_sub_min_eq_abs,abs_sub_comm] dsimp [coordinateUpper,coordinateLower] rw [show max ((C0 : ℝ)/blockLeft C0 R xi H i) ((C0 : ℝ)/blockRight C0 R xi H i)+(H : ℝ)- min ((C0 : ℝ)/blockLeft C0 R xi H i) ((C0 : ℝ)/blockRight C0 R xi H i) = (max ((C0 : ℝ)/blockLeft C0 R xi H i) ((C0 : ℝ)/blockRight C0 R xi H i)- min ((C0 : ℝ)/blockLeft C0 R xi H i) ((C0 : ℝ)/blockRight C0 R xi H i))+(H : ℝ) by ring, he] linarith end ErdosLargeHeightBlocks namespace ErdosStoppedArithmetic open ErdosCommonMInterval ErdosAlignedProgression theorem common_slice_mem_parameter (Y q p : ℕ) (r : ℚ) (R T : ℝ) (m : ℤ) (hq : 0 < q) (hR : 0 < R) (hRp : R ≤ (p : ℝ)) (hpT : (p : ℝ) ≤ T) (hm : (m : ℝ) ∈ interval r.num r.den Y q R T) (hdiv : (r.den : ℤ) ∣ r.num+((p*q : ℕ) : ℤ)*m) : m ∈ parameterInterval Y r (p*q) := by have hp : 0 < p := by exact_mod_cast hR.trans_le hRp have hdiv' : (r.den : ℤ) ∣ r.num+(p : ℤ)*q*m := by simpa only [Nat.cast_mul] using hdiv obtain ⟨n,hn,hY,he⟩ := integer_decode r.den_pos hq hR hRp hpT hm hdiv' apply (mem_parameterInterval Y r (p*q) (Nat.mul_pos hp hq) m).mpr have hD : (0 : ℤ) ≤ r.den := Int.natCast_nonneg _ have hn' : (1 : ℤ) ≤ n := by exact_mod_cast hn have hY' : (n : ℤ) ≤ Y := by exact_mod_cast hY have hlo := mul_le_mul_of_nonneg_left hn' hD have hhi := mul_le_mul_of_nonneg_left hY' hD simp only [mul_one,he] at hlo hhi simpa only [Nat.cast_mul] using And.intro hlo hhi theorem common_slice_endpoint_height (w Cs P R T : ℝ) (A : ℤ) (D Y q : ℕ) (hw : 1 ≤ w) (hP : 1 < P) (hDw : (D : ℝ) ≤ w^Cs) (hY : 0 < Y) (hq : 0 < q) (hR : 0 < R) (hA : |(A : ℝ)| ≤ (Y : ℝ)*w^(Cs+2)) (hne : (interval A D Y q R T).Nonempty) : max |lower A q R T| |upper A D Y q R T| ≤ 2*w^(Cs+2)*P^(lengthExponent P Y R q) := by have hY' : (0 : ℝ) < Y := by exact_mod_cast hY have hq' : (0 : ℝ) < q := by exact_mod_cast hq have hD' : (0 : ℝ) ≤ D := Nat.cast_nonneg _ have hEnds : lower A q R T ≤ upper A D Y q R T := Set.nonempty_Icc.mp hne have hLo := closed_height hD' hY' hq' hR hA (show lower A q R T ∈ interval A D Y q R T from ⟨le_rfl,hEnds⟩) have hHi := closed_height hD' hY' hq' hR hA (show upper A D Y q R T ∈ interval A D Y q R T from ⟨hEnds,le_rfl⟩) have hDC : (D : ℝ) ≤ w^(Cs+2) := hDw.trans (Real.rpow_le_rpow_of_exponent_le hw (by linarith)) have hJ : 0 ≤ referenceLength Y R q := (div_pos hY' (mul_pos hR hq')).le have hBound := mul_le_mul_of_nonneg_right (show (D : ℝ)+w^(Cs+2) ≤ 2*w^(Cs+2) by linarith) hJ rw [reference_rpow hP hY' hR hq'] exact (max_le hLo hHi).trans (by simpa only [mul_assoc] using hBound) end ErdosStoppedArithmetic end Erdos970 end OAI