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8769036 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 | import OAI.NumberTheory.TwoPoint.Halasz.HalaszNormalizedStep
namespace OAI
/-! The geometric defect in the classical Vinogradov iteration. -/
namespace TwoPointCorrelations
def halaszTriangularDegree (k : ℕ) : ℕ := k*(k-1)/2
def halaszTotalDegree (k : ℕ) : ℕ := k+halaszTriangularDegree k
noncomputable def halaszClassicalDefect (k n : ℕ) : ℝ :=
(halaszTriangularDegree k:ℝ)*(1-1/(k:ℝ))^n
noncomputable def halaszClassicalExponent (k n : ℕ) : ℝ :=
2*(((n+1)*k:ℕ):ℝ)-(halaszTotalDegree k:ℝ)+halaszClassicalDefect k n
lemma halasz_triangular_degree_cast {k : ℕ} (hk : 0<k) :
(halaszTriangularDegree k:ℝ)=(k:ℝ)*(k-1)/2 := by
have h := Nat.mul_div_cancel' (Nat.two_dvd_mul_sub_one k)
have hR : (2:ℝ)*(halaszTriangularDegree k:ℝ)=(k:ℝ)*((k-1:ℕ):ℝ) := by
exact_mod_cast h
rw [Nat.cast_sub hk,Nat.cast_one] at hR
linarith
lemma halasz_total_degree_cast {k : ℕ} (hk : 0<k) :
(halaszTotalDegree k:ℝ)=(k:ℝ)*(k+1)/2 := by
simp only [halaszTotalDegree,Nat.cast_add,halasz_triangular_degree_cast hk]
ring
lemma halasz_classical_ratio_bounds {k : ℕ} (hk : 0<k) :
0≤1-1/(k:ℝ) ∧ 1-1/(k:ℝ)≤1 := by
have hk1 : (1:ℝ)≤k := by exact_mod_cast hk
have hk0 : (0:ℝ)<k := by exact_mod_cast hk
have hi : 1/(k:ℝ)≤1 := (div_le_one hk0).mpr hk1
constructor <;> linarith [one_div_pos.mpr hk0]
lemma halasz_classical_defect_bounds {k : ℕ} (hk : 0<k) (n : ℕ) :
0≤halaszClassicalDefect k n ∧
halaszClassicalDefect k n≤(halaszTriangularDegree k:ℝ) := by
have hr := halasz_classical_ratio_bounds hk
constructor
· exact mul_nonneg (Nat.cast_nonneg _) (pow_nonneg hr.1 _)
· exact mul_le_of_le_one_right (Nat.cast_nonneg _)
(pow_le_one₀ hr.1 hr.2)
lemma halasz_classical_defect_successor (k n : ℕ) :
halaszClassicalDefect k (n+1)=(1-1/(k:ℝ))*halaszClassicalDefect k n := by
simp only [halaszClassicalDefect,pow_succ]
ring
lemma halasz_classical_exponent_zero (k : ℕ) : halaszClassicalExponent k 0=k := by
simp only [halaszClassicalExponent,halaszClassicalDefect,pow_zero,mul_one,
zero_add,one_mul,halaszTotalDegree,Nat.cast_add]
ring
lemma halasz_classical_exponent_successor (k n : ℕ) :
halaszClassicalExponent k (n+1)=halaszClassicalExponent k n+2*k-
halaszClassicalDefect k n/(k:ℝ) := by
simp only [halaszClassicalExponent,halasz_classical_defect_successor,
Nat.cast_mul,Nat.cast_add,Nat.cast_one]
ring
lemma halasz_classical_exponent_bounds {k : ℕ} (hk : 0<k) (n : ℕ) :
(k:ℝ)≤halaszClassicalExponent k n ∧
halaszClassicalExponent k n≤(halaszStepExponent ((n+1)*k) k:ℝ) := by
have hk0 : (0:ℝ)<k := by exact_mod_cast hk
have hd (n : ℕ) : halaszClassicalDefect k n/(k:ℝ)≤k := by
apply (div_le_iff₀ hk0).mpr
have hh := (halasz_classical_defect_bounds hk n).2
rw [halasz_triangular_degree_cast hk] at hh
nlinarith
constructor
· induction n with
| zero => exact (halasz_classical_exponent_zero k).ge
| succ n ih =>
rw [halasz_classical_exponent_successor]
linarith [hd n]
· have hh := (halasz_classical_defect_bounds hk n).2
simp only [halaszClassicalExponent,halaszTotalDegree,Nat.cast_add,
halaszStepExponent,Nat.cast_mul,Nat.cast_ofNat,Nat.cast_one]
change 2*(((n:ℝ)+1)*k)-((k:ℝ)+(halaszTriangularDegree k:ℝ))+
halaszClassicalDefect k n ≤ 2*(((n:ℝ)+1)*k)+(halaszTriangularDegree k:ℝ)
linarith [show (0:ℝ)≤(halaszTriangularDegree k:ℝ) from Nat.cast_nonneg _]
lemma halasz_classical_exponent_normalization {k : ℕ} (hk : 0<k) (n : ℕ) :
(k:ℝ)+halaszClassicalExponent k n +
((halaszStepExponent ((n+1)*k) k:ℝ)-halaszClassicalExponent k n)/(k:ℝ) =
halaszClassicalExponent k (n+1) := by
have hk0 : (k:ℝ)≠0 := by exact_mod_cast hk.ne'
have ha : (halaszStepExponent ((n+1)*k) k:ℝ) =
2*(((n+1)*k:ℕ):ℝ)+(halaszTriangularDegree k:ℝ) := by
simp only [halaszStepExponent,halaszTriangularDegree,Nat.cast_add,Nat.cast_mul,
Nat.cast_ofNat]
rw [ha]
simp only [halaszClassicalExponent,halasz_classical_defect_successor,
Nat.cast_add,Nat.cast_mul,Nat.cast_one]
rw [halasz_total_degree_cast hk,halasz_triangular_degree_cast hk]
field_simp
ring
end TwoPointCorrelations
end OAI
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