| import OAI.NumberTheory.TwoPoint.Halasz.HalaszNormalizedStep |
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| namespace OAI |
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| /-! The geometric defect in the classical Vinogradov iteration. -/ |
| namespace TwoPointCorrelations |
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| def halaszTriangularDegree (k : ℕ) : ℕ := k*(k-1)/2 |
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| def halaszTotalDegree (k : ℕ) : ℕ := k+halaszTriangularDegree k |
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| noncomputable def halaszClassicalDefect (k n : ℕ) : ℝ := |
| (halaszTriangularDegree k:ℝ)*(1-1/(k:ℝ))^n |
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| noncomputable def halaszClassicalExponent (k n : ℕ) : ℝ := |
| 2*(((n+1)*k:ℕ):ℝ)-(halaszTotalDegree k:ℝ)+halaszClassicalDefect k n |
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| 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 |
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|
| 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 |
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| 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] |
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| 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) |
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| 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 |
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| 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 |
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| 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 |
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|
| 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 _] |
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| 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 |
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|
| end TwoPointCorrelations |
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| end OAI |
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