| import OAI.NumberTheory.PrimeGaps.CharacterErrors |
|
|
| namespace OAI |
|
|
| namespace LargePrimeGaps |
|
|
| open Filter |
|
|
| open Set Filter MeasureTheory |
|
|
| open scoped Topology ContDiff |
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|
| open Asymptotics |
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|
| open Asymptotics |
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|
| open Asymptotics |
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|
| open scoped Classical |
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|
| open scoped ContDiff |
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|
| open Topology |
|
|
| open scoped Convolution ContDiff Pointwise |
|
|
| open scoped ComplexConjugate |
|
|
| theorem symmetric_kernel_quadratic_le {ι : Type*} (s : Finset ι) (a : ι→ℝ) |
| (w : ι→ι→ℝ) (hw : ∀ i∈s,∀ j∈s,0≤w i j) |
| (hs : ∀ i∈s,∀ j∈s,w i j=w j i) : |
| (∑ i∈s,∑ j∈s,a i*a j*w i j)≤∑ i∈s,a i^2*(∑ j∈s,w i j) := by |
| have hh : (∑ i∈s,∑ j∈s,a i*a j*w i j)≤ |
| ∑ i∈s,∑ j∈s,(a i^2*w i j+a j^2*w i j)/2 := by |
| apply Finset.sum_le_sum |
| intro i hi |
| apply Finset.sum_le_sum |
| intro j hj |
| have h := mul_nonneg (sq_nonneg (a i-a j)) (hw i hi j hj) |
| nlinarith |
| have hswap : (∑ i∈s,∑ j∈s,a j^2*w i j)=(∑ i∈s,∑ j∈s,a i^2*w i j) := by |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro i hi |
| apply Finset.sum_congr rfl |
| intro j hj |
| rw [hs j hj i hi] |
| calc |
| _ ≤ _ := hh |
| _ = _ := by |
| simp_rw [←Finset.sum_div,Finset.sum_add_distrib] |
| rw [hswap] |
| simp only [Finset.mul_sum] |
| ring |
|
|
| theorem norm_sum_smul_sq_le_gram {ι E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E] |
| (s : Finset ι) (v : ι→E) (c : ι→ℂ) : |
| ‖∑ i∈s,c i • v i‖^2≤∑ i∈s,‖c i‖^2*(∑ j∈s,‖inner ℂ (v i) (v j)‖) := by |
| calc |
| _ = ∑ i∈s,∑ j∈s,(conj (c i)*(c j*inner ℂ (v i) (v j))).re := by |
| rw [InnerProductSpace.norm_sq_eq_re_inner (𝕜:=ℂ)] |
| simp only [sum_inner,inner_sum,inner_smul_left,inner_smul_right,Finset.mul_sum,map_sum] |
| rw [Finset.sum_comm] |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro j _ |
| change (c j * (conj (c i) * inner ℂ (v i) (v j))).re = _ |
| congr 1 |
| ring |
| _ ≤ ∑ i∈s,∑ j∈s,‖c i‖*‖c j‖*‖inner ℂ (v i) (v j)‖ := by |
| apply Finset.sum_le_sum |
| intro i _ |
| apply Finset.sum_le_sum |
| intro j _ |
| simpa only [norm_mul,Complex.norm_conj,mul_assoc] using |
| Complex.re_le_norm (conj (c i)*(c j*inner ℂ (v i) (v j))) |
| _ ≤ _ := symmetric_kernel_quadratic_le s (fun i => ‖c i‖) |
| (fun i j => ‖inner ℂ (v i) (v j)‖) (fun _ _ _ _ => norm_nonneg _) |
| (fun i _ j _ => by rw [←inner_conj_symm,Complex.norm_conj]) |
| |
| theorem gram_large_sieve {ι E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℂ E] |
| (s : Finset ι) (v : ι→E) {B : ℝ} (hB : 0≤B) |
| (hrows : ∀ i∈s,(∑ j∈s,‖inner ℂ (v i) (v j)‖)≤B) (x : E) : |
| (∑ i∈s,‖inner ℂ (v i) x‖^2)≤B*‖x‖^2 := by |
| let c : ι→ℂ := fun i => inner ℂ (v i) x |
| let y : E := ∑ i∈s,c i • v i |
| let A : ℝ := ∑ i∈s,‖c i‖^2 |
| have hA : 0≤A := Finset.sum_nonneg (fun _ _ => sq_nonneg _) |
| have hy : ‖y‖^2≤B*A := by |
| apply (norm_sum_smul_sq_le_gram s v c).trans |
| calc |
| _ ≤ ∑ i∈s,‖c i‖^2*B := Finset.sum_le_sum |
| (fun i hi => mul_le_mul_of_nonneg_left (hrows i hi) (sq_nonneg _)) |
| _ = B*A := by rw [←Finset.sum_mul,mul_comm] |
| have hxy : inner ℂ x y=(A:ℂ) := by |
| simp only [y,inner_sum,inner_smul_right,A,Complex.ofReal_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| change inner ℂ (v i) x * inner ℂ x (v i)=_ |
| rw [←inner_conj_symm x (v i)] |
| simpa only [Complex.normSq_eq_norm_sq,c] using Complex.mul_conj (inner ℂ (v i) x) |
| have hn := norm_inner_le_norm (𝕜:=ℂ) x y |
| rw [hxy,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg hA] at hn |
| have hn2 : A^2≤‖x‖^2*‖y‖^2 := by |
| nlinarith [norm_nonneg x,norm_nonneg y] |
| have hprod := mul_le_mul_of_nonneg_left hy (sq_nonneg ‖x‖) |
| change A≤B*‖x‖^2 |
| by_cases hA0 : A=0 |
| · rw [hA0] |
| exact mul_nonneg hB (sq_nonneg _) |
| · apply le_of_mul_le_mul_right (a := A) ?_ (lt_of_le_of_ne hA (Ne.symm hA0)) |
| calc |
| A*A = A^2 := by ring |
| _ ≤ _ := hn2.trans hprod |
| _ = (B*‖x‖^2)*A := by ring |
|
|
| noncomputable def bvExp (x : ℝ) : ℂ := Complex.exp (Complex.I*(2*Real.pi*x:ℝ)) |
|
|
| @[simp] theorem bvExp_zero : bvExp 0=1 := by simp [bvExp] |
| |
| @[simp] theorem norm_bvExp (x : ℝ) : ‖bvExp x‖=1 := by simp [bvExp,Complex.norm_exp] |
| |
| theorem bvExp_add (x y : ℝ) : bvExp (x+y)=bvExp x*bvExp y := by |
| unfold bvExp |
| rw [←Complex.exp_add] |
| congr 1 |
| push_cast |
| ring |
|
|
| theorem conj_bvExp (x : ℝ) : conj (bvExp x)=bvExp (-x) := by |
| unfold bvExp |
| rw [←Complex.exp_conj] |
| congr 1 |
| simp only [map_mul, Complex.conj_I, Complex.conj_ofReal] |
| push_cast |
| ring |
|
|
| theorem bvExp_nat_mul (n : ℕ) (x : ℝ) : bvExp ((n:ℝ)*x)=bvExp x^n := by |
| unfold bvExp |
| rw [←Complex.exp_nat_mul] |
| congr 1 |
| push_cast |
| ring |
|
|
| theorem bvExp_sub_one_lower {x : ℝ} (hx : |x|≤1/2) : |
| 4*|x|≤‖bvExp x-1‖ := by |
| rw [bvExp,Complex.norm_exp_I_mul_ofReal_sub_one] |
| have hs : |Real.pi*x|≤Real.pi/2 := by |
| rw [abs_mul,abs_of_pos Real.pi_pos] |
| nlinarith [Real.pi_pos] |
| have h := Real.mul_abs_le_abs_sin hs |
| rw [abs_mul,abs_of_pos Real.pi_pos] at h |
| have hpi : Real.pi≠0 := ne_of_gt Real.pi_pos |
| have he : 2/Real.pi*(Real.pi*|x|)=2*|x| := by field_simp |
| rw [he] at h |
| have hid : 2*Real.pi*x/2=Real.pi*x := by ring |
| rw [hid,Real.norm_eq_abs,abs_mul,abs_of_pos (by norm_num : (0:ℝ)<2)] |
| linarith |
|
|
| theorem exponential_sum_bound (N : ℕ) {x : ℝ} (hx : |x|≤1/2) (hx0 : x≠0) : |
| ‖∑ n∈Finset.range N,bvExp ((n:ℝ)*x)‖≤1/(2*|x|) := by |
| have ha : 0 < |x| := abs_pos.mpr hx0 |
| have hden := bvExp_sub_one_lower hx |
| have hn : 0<‖bvExp x-1‖ := lt_of_lt_of_le (by positivity : 0<4*|x|) hden |
| have he := geom_sum_mul (bvExp x) N |
| have hnorm := congrArg norm he |
| rw [norm_mul] at hnorm |
| have hu : ‖bvExp x^N-1‖≤2 := by |
| calc |
| _ ≤ ‖bvExp x^N‖+‖(1:ℂ)‖ := norm_sub_le _ _ |
| _ = 2 := by simp [norm_pow]; norm_num |
| have htimes : ‖∑ n∈Finset.range N,bvExp x^n‖*(4*|x|)≤2 := by |
| exact (mul_le_mul_of_nonneg_left hden (norm_nonneg _)).trans (hnorm.trans_le hu) |
| simp_rw [bvExp_nat_mul] |
| apply (le_div_iff₀ (by positivity : (0:ℝ)<2*|x|)).mpr |
| nlinarith |
|
|
| theorem reciprocal_sum_le_harmonic {ι : Type*} (s : Finset ι) (t : ι→ℝ) (D : ℕ) |
| (ht : ∀ i∈s,1≤t i ∧ t i≤D) |
| (hsep : ∀ i∈s,∀ j∈s,i≠j→1≤|t i-t j|) : |
| (∑ i∈s,1/t i)≤(harmonic D:ℝ) := by |
| classical |
| let r : ι→ℕ := fun i => ⌊t i⌋₊ |
| have hr : ∀ i∈s,r i∈Finset.Icc 1 D := by |
| intro i hi |
| exact Finset.mem_Icc.mpr ⟨Nat.le_floor (by simpa using (ht i hi).1), |
| Nat.floor_le_of_le (ht i hi).2⟩ |
| have hinj : Set.InjOn r s := by |
| intro i hi j hj he |
| by_contra hij |
| have h := hsep i hi j hj hij |
| have hil := Nat.floor_le (le_trans (by norm_num : (0:ℝ)≤1) (ht i hi).1) |
| have hjl := Nat.floor_le (le_trans (by norm_num : (0:ℝ)≤1) (ht j hj).1) |
| have hiu := Nat.lt_floor_add_one (t i) |
| have hju := Nat.lt_floor_add_one (t j) |
| change ⌊t i⌋₊=⌊t j⌋₊ at he |
| rw [he] at hil hiu |
| rcases le_abs.mp h with h|h <;> linarith |
| calc |
| _ ≤ ∑ i∈s,1/(r i:ℝ) := by |
| apply Finset.sum_le_sum |
| intro i hi |
| apply one_div_le_one_div_of_le |
| · exact_mod_cast (Finset.mem_Icc.mp (hr i hi)).1 |
| · exact Nat.floor_le (le_trans (by norm_num : (0:ℝ)≤1) (ht i hi).1) |
| _ = ∑ k∈s.image r,1/(k:ℝ) := (Finset.sum_image (f := fun k : ℕ => (1:ℝ)/k) hinj).symm |
| _ ≤ ∑ k∈Finset.Icc 1 D,1/(k:ℝ) := |
| Finset.sum_le_sum_of_subset_of_nonneg (Finset.image_subset_iff.mpr hr) |
| (fun _ _ _ => by positivity) |
| _ = _ := by simp [harmonic_eq_sum_Icc] |
| |
| theorem reciprocal_sum_le_scaled_harmonic {ι : Type*} (s : Finset ι) (t : ι→ℝ) |
| {D : ℕ} (hD : 0<D) (ht : ∀ i∈s,1/(D:ℝ)≤t i ∧ t i≤1) |
| (hsep : ∀ i∈s,∀ j∈s,i≠j→1/(D:ℝ)≤|t i-t j|) : |
| (∑ i∈s,1/t i)≤(D:ℝ)*(harmonic D:ℝ) := by |
| have hDr : 0<(D:ℝ) := by exact_mod_cast hD |
| have hh := reciprocal_sum_le_harmonic s (fun i => (D:ℝ)*t i) D |
| (fun i hi => ⟨by have h := mul_le_mul_of_nonneg_left (ht i hi).1 hDr.le |
| field_simp at h |
| nlinarith, |
| by nlinarith [(ht i hi).2]⟩) |
| (fun i hi j hj hij => by |
| rw [←mul_sub,abs_mul,abs_of_pos hDr] |
| have h := (div_le_iff₀ hDr).mp (hsep i hi j hj hij) |
| nlinarith) |
| have hm := mul_le_mul_of_nonneg_left hh hDr.le |
| rw [Finset.mul_sum] at hm |
| calc |
| _ = ∑ i∈s,(D:ℝ)*(1/((D:ℝ)*t i)) := by |
| apply Finset.sum_congr rfl |
| intro i _ |
| field_simp |
| _ ≤ _ := hm |
|
|
| @[simp] theorem bvExp_int (m : ℤ) : bvExp m=1 := by |
| unfold bvExp |
| have he : Complex.I * ((2*Real.pi*(m:ℝ):ℝ):ℂ)=(m:ℂ)*(2*Real.pi*Complex.I) := by |
| push_cast |
| ring |
| rw [he,Complex.exp_int_mul,Complex.exp_two_pi_mul_I,one_zpow] |
|
|
| theorem bvExp_sub_int (x : ℝ) (m : ℤ) : bvExp (x-m)=bvExp x := by |
| rw [sub_eq_add_neg,bvExp_add] |
| have h : bvExp (-(m:ℝ))=1 := by simpa using bvExp_int (-m) |
| rw [h,mul_one] |
| |
| noncomputable def bvCentered (x : ℝ) : ℝ := x-(⌊x+1/2⌋:ℤ) |
|
|
| theorem abs_bvCentered_le (x : ℝ) : |bvCentered x|≤1/2 := by |
| have hl := Int.floor_le (x+1/2) |
| have hu := Int.lt_floor_add_one (x+1/2) |
| dsimp [bvCentered] |
| exact abs_le.mpr ⟨by linarith,by linarith⟩ |
|
|
| theorem bvExp_centered (x : ℝ) : bvExp (bvCentered x)=bvExp x := bvExp_sub_int _ _ |
|
|
| theorem exponential_sum_centered (N : ℕ) (x : ℝ) : |
| (∑ n∈Finset.range N,bvExp ((n:ℝ)*bvCentered x))= |
| ∑ n∈Finset.range N,bvExp ((n:ℝ)*x) := by |
| simp_rw [bvExp_nat_mul,bvExp_centered] |
| |
| theorem centered_difference_separated {ι : Type*} (s : Finset ι) (θ : ι→ℝ) {D : ℕ} |
| (hsep : ∀ i∈s,∀ j∈s,i≠j→∀ m:ℤ,1/(D:ℝ)≤|θ i-θ j-m|) |
| (a : ℝ) {i j : ι} (hi : i∈s) (hj : j∈s) (hij : i≠j) : |
| 1/(D:ℝ)≤|bvCentered (θ i-a)-bvCentered (θ j-a)| := by |
| have h := hsep i hi j hj hij (⌊θ i-a+1/2⌋-⌊θ j-a+1/2⌋) |
| dsimp [bvCentered] |
| convert h using 2 |
| push_cast |
| ring |
|
|
| theorem exponential_kernel_row_bound {ι : Type*} (s : Finset ι) (θ : ι→ℝ) |
| {D : ℕ} (hD : 0<D) |
| (hsep : ∀ i∈s,∀ j∈s,i≠j→∀ m:ℤ,1/(D:ℝ)≤|θ i-θ j-m|) |
| (N : ℕ) {i : ι} (hi : i∈s) : |
| (∑ j∈s,‖∑ n∈Finset.range N,bvExp ((n:ℝ)*(θ j-θ i))‖)≤ |
| (N:ℝ)+(D:ℝ)*(harmonic D:ℝ) := by |
| classical |
| let t := s.erase i |
| let r := fun j => bvCentered (θ j-θ i) |
| have hDr : 0<(D:ℝ) := by exact_mod_cast hD |
| have hr : ∀ j∈t,1/(D:ℝ)≤|r j| := by |
| intro j hj |
| rcases Finset.mem_erase.mp hj with ⟨hji,hjs⟩ |
| exact hsep j hjs i hi hji ⌊θ j-θ i+1/2⌋ |
| have hn : ∀ j∈t,r j≠0 := by |
| intro j hj hj0 |
| have h := hr j hj |
| rw [hj0,abs_zero] at h |
| exact (not_le_of_gt (one_div_pos.mpr hDr)) h |
| have hpair : ∀ j∈t,∀ k∈t,j≠k→1/(D:ℝ)≤|r j-r k| := by |
| intro j hj k hk hjk |
| exact centered_difference_separated s θ hsep (θ i) |
| (Finset.mem_of_mem_erase hj) (Finset.mem_of_mem_erase hk) hjk |
| have hhalf : ∀ b:Bool, |
| (∑ j∈t.filter (fun j => decide (0≤r j)=b),1/|r j|)≤(D:ℝ)*(harmonic D:ℝ) := by |
| intro b |
| apply reciprocal_sum_le_scaled_harmonic _ _ hD |
| · intro j hj |
| exact ⟨hr j ((Finset.mem_filter.mp hj).1),(abs_bvCentered_le _).trans (by norm_num)⟩ |
| · intro j hj k hk hjk |
| have h := hpair j ((Finset.mem_filter.mp hj).1) k ((Finset.mem_filter.mp hk).1) hjk |
| have hjb := (Finset.mem_filter.mp hj).2 |
| have hkb := (Finset.mem_filter.mp hk).2 |
| cases b < |
| · simpa only [abs_of_neg hjb,abs_of_neg hkb,neg_sub_neg,abs_sub_comm] using h |
| · simpa only [abs_of_nonneg hjb,abs_of_nonneg hkb] using h |
| have hrec : (∑ j∈t,1/|r j|)≤2*(D:ℝ)*(harmonic D:ℝ) := by |
| have hp := hhalf true |
| have hm := hhalf false |
| simp only [decide_eq_true_eq,decide_eq_false_iff_not] at hp hm |
| have he := Finset.sum_filter_add_sum_filter_not t (fun j => 0≤r j) (fun j => 1/|r j|) |
| linarith |
| have hoff : (∑ j∈t,‖∑ n∈Finset.range N,bvExp ((n:ℝ)*(θ j-θ i))‖)≤ |
| (D:ℝ)*(harmonic D:ℝ) := by |
| calc |
| _ ≤ ∑ j∈t,1/(2*|r j|) := by |
| apply Finset.sum_le_sum |
| intro j hj |
| rw [←exponential_sum_centered] |
| exact exponential_sum_bound N (abs_bvCentered_le _) (hn j hj) |
| _ = (∑ j∈t,1/|r j|)/2 := by |
| rw [Finset.sum_div] |
| apply Finset.sum_congr rfl |
| intro j _ |
| ring |
| _ ≤ _ := by linarith |
| rw [←Finset.sum_erase_add _ _ hi] |
| have hd : ‖∑ n∈Finset.range N,bvExp ((n:ℝ)*(θ i-θ i))‖=(N:ℝ) := by simp |
| rw [hd] |
| linarith |
|
|
| noncomputable def bvExponentialVector (N : ℕ) (θ : ℝ) : EuclideanSpace ℂ (Fin N) := |
| WithLp.toLp 2 (fun n => bvExp (-((n:ℕ):ℝ)*θ)) |
|
|
| theorem inner_bvExponentialVector (N : ℕ) (θ η : ℝ) : |
| inner ℂ (bvExponentialVector N θ) (bvExponentialVector N η)= |
| ∑ n∈Finset.range N,bvExp ((n:ℝ)*(θ-η)) := by |
| simp only [PiLp.inner_apply,RCLike.inner_apply,bvExponentialVector] |
| simp_rw [mul_comm (bvExp _)] |
| simp_rw [conj_bvExp,←bvExp_add] |
| have he : ∀ n:Fin N,-(-((n:ℕ):ℝ)*θ)+(-((n:ℕ):ℝ)*η)=((n:ℕ):ℝ)*(θ-η) := by |
| intro n |
| ring |
| simp_rw [he] |
| exact Fin.sum_univ_eq_sum_range (fun n : ℕ => bvExp ((n:ℝ)*(θ-η))) N |
|
|
| theorem inner_bvExponentialVector_coefficients (N : ℕ) (θ : ℝ) (a : Fin N→ℂ) : |
| inner ℂ (bvExponentialVector N θ) (WithLp.toLp 2 a)= |
| ∑ n:Fin N,a n*bvExp (((n:ℕ):ℝ)*θ) := by |
| simp only [PiLp.inner_apply,RCLike.inner_apply,bvExponentialVector] |
| apply Finset.sum_congr rfl |
| intro n _ |
| rw [conj_bvExp] |
| congr 2 |
| ring |
|
|
| theorem additive_large_sieve {ι : Type*} (s : Finset ι) (θ : ι→ℝ) |
| {D : ℕ} (hD : 0<D) |
| (hsep : ∀ i∈s,∀ j∈s,i≠j→∀ m:ℤ,1/(D:ℝ)≤|θ i-θ j-m|) |
| (N : ℕ) (a : Fin N→ℂ) : |
| (∑ i∈s,‖∑ n:Fin N,a n*bvExp (((n:ℕ):ℝ)*θ i)‖^2)≤ |
| ((N:ℝ)+(D:ℝ)*(harmonic D:ℝ))*(∑ n:Fin N,‖a n‖^2) := by |
| have hH : 0≤(harmonic D:ℝ) := by |
| rw [harmonic_eq_sum_Icc,Rat.cast_sum] |
| exact Finset.sum_nonneg (fun _ _ => by simp only [Rat.cast_inv,Rat.cast_natCast] |
| have hB : 0≤(N:ℝ)+(D:ℝ)*(harmonic D:ℝ) := by positivity |
| have hr : ∀ i∈s,(∑ j∈s,‖inner ℂ (bvExponentialVector N (θ i)) |
| (bvExponentialVector N (θ j))‖)≤(N:ℝ)+(D:ℝ)*(harmonic D:ℝ) := by |
| intro i hi |
| simp_rw [inner_bvExponentialVector] |
| have hh := exponential_kernel_row_bound s (fun i => -θ i) hD |
| (fun j hj k hk hjk m => by |
| have h := hsep j hj k hk hjk (-m) |
| have he : -θ j- -θ k-(m:ℝ)=-(θ j-θ k-(-m:ℤ)) := by |
| push_cast |
| ring |
| rw [he,abs_neg] |
| exact h) N hi |
| convert hh using 1 |
| apply Finset.sum_congr rfl |
| intro j _ |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro n _ |
| congr 1 |
| ring |
| have h := gram_large_sieve s (fun i => bvExponentialVector N (θ i)) hB hr (WithLp.toLp 2 a) |
| simpa only [inner_bvExponentialVector_coefficients,EuclideanSpace.norm_sq_eq,WithLp.ofLp_toLp] using h |
|
|
| theorem rat_abs_lower (r : ℚ) (hr : r≠0) : 1/(r.den:ℝ)≤|(r:ℝ)| := by |
| have hn : r.num≠0 := fun h => hr (Rat.zero_of_num_zero h) |
| have ha : (1:ℝ)≤|(r.num:ℝ)| := by |
| exact_mod_cast Int.one_le_abs hn |
| rw [Rat.cast_def,abs_div,abs_of_pos (by exact_mod_cast r.den_pos : (0:ℝ)<r.den)] |
| exact div_le_div_of_nonneg_right ha (Nat.cast_nonneg _) |
|
|
| theorem rational_frequency_separation {r t : ℚ} {Q : ℕ} |
| (hr : 0≤r ∧ r<1) (ht : 0≤t ∧ t<1) (hrt : r≠t) |
| (hrQ : r.den≤Q) (htQ : t.den≤Q) (m : ℤ) : |
| 1/((Q^2:ℕ):ℝ)≤|(r:ℝ)-(t:ℝ)-m| := by |
| have hu0 : r-t-(m:ℚ)≠0 := by |
| intro he |
| have hm0 : m=0 := by |
| have hlo : (-1:ℚ)<(m:ℚ) := by linarith [hr.1,ht.2] |
| have hup : (m:ℚ)<1 := by linarith [ht.1,hr.2] |
| have h1 : (-1:ℤ) < m := by exact_mod_cast hlo |
| have h2 : m<1 := by exact_mod_cast hup |
| omega |
| simp only [hm0,Int.cast_zero,sub_zero,sub_eq_zero] at he |
| exact hrt he |
| have hd : (r-t-(m:ℚ)).den≤Q^2 := by |
| rw [Rat.sub_intCast_den] |
| calc |
| _ ≤ r.den*t.den := Nat.le_of_dvd (Nat.mul_pos r.den_pos t.den_pos) (Rat.sub_den_dvd _ _) |
| _ ≤ Q*Q := Nat.mul_le_mul hrQ htQ |
| _ = _ := by ring |
| have hl := rat_abs_lower (r-t-(m:ℚ)) hu0 |
| push_cast at hl |
| exact (one_div_le_one_div_of_le |
| (by exact_mod_cast (r-t-(m:ℚ)).den_pos : (0:ℝ)<(r-t-(m:ℚ)).den) |
| (by exact_mod_cast hd)).trans hl |
|
|
| theorem rational_additive_large_sieve (s : Finset ℚ) {Q : ℕ} (hQ : 0<Q) |
| (hs : ∀ r∈s,0≤r ∧ r<1 ∧ r.den≤Q) (N : ℕ) (a : Fin N→ℂ) : |
| (∑ r∈s,‖∑ n:Fin N,a n*bvExp (((n:ℕ):ℝ)*(r:ℝ))‖^2)≤ |
| ((N:ℝ)+(Q:ℝ)^2*(harmonic (Q^2):ℝ))*(∑ n:Fin N,‖a n‖^2) := by |
| have h := additive_large_sieve s (fun r => (r:ℝ)) (pow_pos hQ 2) |
| (fun r hr t ht hrt m => rational_frequency_separation |
| ⟨(hs r hr).1,(hs r hr).2.1⟩ ⟨(hs t ht).1,(hs t ht).2.1⟩ hrt |
| (hs r hr).2.2 (hs t ht).2.2 m) N a |
| simpa only [Nat.cast_pow] using h |
|
|
| end LargePrimeGaps |
|
|
| end OAI |
|
|
