| import OAI.NumberTheory.Ostmann.Arithmetic.FrequencyTreeSum |
| import OAI.NumberTheory.Ostmann.Characters.SignedFrequencySum |
|
|
| namespace OAI |
|
|
| /-! # The full arithmetic frequency-tree budget -/ |
|
|
| namespace Ostmann |
|
|
| open scoped BigOperators Classical |
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|
| noncomputable def pairedFrequencyKernel (S : Finset ℤ) (D : ℝ) |
| (root left right : S × S) : ℝ := |
| 2 * D ^ 2 * |
| (((reducedFrequency ((left.1 : ℤ).natAbs.gcd (right.1 : ℤ).natAbs) (root.1 : ℤ).natAbs).lcm |
| (reducedFrequency ((left.2 : ℤ).natAbs.gcd (right.2 : ℤ).natAbs) (root.2 : ℤ).natAbs) : ℕ) : ℝ)⁻¹ |
|
|
| theorem pairedFrequencyKernel_sum_le (S : Finset ℤ) (N : ℕ) (D : ℝ) (hD : 0 ≤ D) |
| (hS : ∀ s ∈ S, s ≠ 0 ∧ s.natAbs ≤ N) |
| (hdiv : ∀ v ∈ S, ∀ w ∈ S, ((v.natAbs.gcd w.natAbs).divisors.card : ℝ) ≤ D) |
| (left right : S × S) : |
| (∑ root : S × S, pairedFrequencyKernel S D root left right) ≤ |
| 8 * D ^ 4 * (1 + Real.log N) ^ 3 := by |
| let g := (left.1 : ℤ).natAbs.gcd (right.1 : ℤ).natAbs |
| let h := (left.2 : ℤ).natAbs.gcd (right.2 : ℤ).natAbs |
| have hg : g ≠ 0 := Nat.ne_of_gt (Nat.gcd_pos_of_pos_left _ |
| (Int.natAbs_pos.mpr (hS _ left.1.property).1)) |
| have hh : h ≠ 0 := Nat.ne_of_gt (Nat.gcd_pos_of_pos_left _ |
| (Int.natAbs_pos.mpr (hS _ left.2.property).1)) |
| have hgd : (g.divisors.card : ℝ) ≤ D := hdiv _ left.1.property _ right.1.property |
| have hhd : (h.divisors.card : ℝ) ≤ D := hdiv _ left.2.property _ right.2.property |
| have hpair : |
| (∑ s : S, ∑ t : S, |
| (((reducedFrequency g (s : ℤ).natAbs).lcm (reducedFrequency h (t : ℤ).natAbs) : ℕ) : ℝ)⁻¹) = |
| ∑ s ∈ S, ∑ t ∈ S, |
| (((reducedFrequency g s.natAbs).lcm (reducedFrequency h t.natAbs) : ℕ) : ℝ)⁻¹ := by |
| rw [Finset.sum_coe_sort S (fun s : ℤ => ∑ t : S, |
| (((reducedFrequency g s.natAbs).lcm (reducedFrequency h (t : ℤ).natAbs) : ℕ) : ℝ)⁻¹)] |
| exact Finset.sum_congr rfl fun s _ => Finset.sum_coe_sort S (fun t : ℤ => |
| (((reducedFrequency g s.natAbs).lcm (reducedFrequency h t.natAbs) : ℕ) : ℝ)⁻¹) |
| calc |
| _ = 2 * D ^ 2 * (∑ s : S, ∑ t : S, |
| (((reducedFrequency g (s : ℤ).natAbs).lcm (reducedFrequency h (t : ℤ).natAbs) : ℕ) : ℝ)⁻¹) := by |
| simp only [pairedFrequencyKernel, Fintype.sum_prod_type, ← Finset.mul_sum, g, h] |
| _ ≤ 2 * D ^ 2 * (4 * (g.divisors.card : ℝ) * h.divisors.card * (1 + Real.log N) ^ 3) := by |
| rw [hpair] |
| exact mul_le_mul_of_nonneg_left (signed_reducedFrequency_lcm_sum_le S S N g h hS hS hg hh) |
| (by positivity) |
| _ ≤ 2 * D ^ 2 * (4 * D * D * (1 + Real.log N) ^ 3) := by |
| apply mul_le_mul_of_nonneg_left _ (by positivity) |
| have hlog : 0 ≤ (1 + Real.log N) ^ 3 := by |
| have hbase := Real.log_natCast_nonneg N |
| positivity |
| exact mul_le_mul_of_nonneg_right |
| (mul_le_mul (mul_le_mul_of_nonneg_left hgd (by norm_num)) hhd |
| (Nat.cast_nonneg _) (by positivity)) hlog |
| _ = _ := by ring |
|
|
| noncomputable def pairedFrequencyLeaf (S : Finset ℤ) (V : ℕ) (x : S × S) : ℝ := |
| if (x.1 : ℤ).natAbs ≤ V ∧ (x.2 : ℤ).natAbs ≤ V then 1 else 0 |
|
|
| theorem pairedFrequencyLeaf_sum_le (S : Finset ℤ) (V : ℕ) |
| (hS : ∀ s ∈ S, s ≠ 0) : |
| (∑ x : S × S, pairedFrequencyLeaf S V x) ≤ (2 * (V : ℝ)) ^ 2 := by |
| let T := S.filter fun s => s.natAbs ≤ V |
| have hT : ∀ s ∈ T, s ≠ 0 ∧ s.natAbs ≤ V := by |
| intro s hs |
| exact ⟨hS s (Finset.mem_filter.mp hs).1, (Finset.mem_filter.mp hs).2⟩ |
| have hcard : (T.card : ℝ) ≤ 2 * V := by |
| have hbound := sum_signed_magnitudes_le T V hT (fun _ => (1 : ℝ)) (by intros; norm_num) |
| simpa using hbound |
| have hsingle : (∑ s : S, if (s : ℤ).natAbs ≤ V then (1 : ℝ) else 0) = T.card := by |
| rw [Finset.sum_coe_sort S (fun s : ℤ => if s.natAbs ≤ V then (1 : ℝ) else 0)] |
| simp only [← Finset.sum_filter, Finset.sum_const, nsmul_eq_mul, mul_one, T] |
| have hterm (x y : S) : pairedFrequencyLeaf S V (x, y) = |
| (if (x : ℤ).natAbs ≤ V then (1 : ℝ) else 0) * |
| (if (y : ℤ).natAbs ≤ V then (1 : ℝ) else 0) := by |
| unfold pairedFrequencyLeaf |
| split_ifs <;> simp_all |
| calc |
| _ = (T.card : ℝ) ^ 2 := by |
| rw [Fintype.sum_prod_type, pow_two, ← hsingle, Finset.sum_mul_sum] |
| simp only [hterm] |
| _ ≤ _ := pow_le_pow_left₀ (Nat.cast_nonneg _) hcard 2 |
|
|
| / |
| The exponent `2^n-1` is the exact number of internal nodes. -/ |
| theorem arithmetic_frequency_tree_sum_le (S : Finset ℤ) (N V : ℕ) (D : ℝ) |
| (hD : 0 ≤ D) (hS : ∀ s ∈ S, s ≠ 0 ∧ s.natAbs ≤ N) |
| (hdiv : ∀ v ∈ S, ∀ w ∈ S, ((v.natAbs.gcd w.natAbs).divisors.card : ℝ) ≤ D) |
| (n : ℕ) : |
| (∑ x : FrequencyTree (S × S) n, |
| frequencyTreeWeight (pairedFrequencyKernel S D) (pairedFrequencyLeaf S V) n x) ≤ |
| (8 * D ^ 4 * (1 + Real.log N) ^ 3) ^ (2 ^ n - 1) * (2 * (V : ℝ)) ^ (2 * 2 ^ n) := by |
| have hC : 0 ≤ 8 * D ^ 4 * (1 + Real.log N) ^ 3 := by |
| have hlog := Real.log_natCast_nonneg N |
| positivity |
| have hmain := frequencyTree_sum_le (pairedFrequencyKernel S D) (pairedFrequencyLeaf S V) |
| (8 * D ^ 4 * (1 + Real.log N) ^ 3) (by intros; unfold pairedFrequencyKernel; positivity) |
| (by intros; unfold pairedFrequencyLeaf; split_ifs <;> norm_num) hC |
| (pairedFrequencyKernel_sum_le S N D hD hS hdiv) n |
| apply hmain.trans |
| have hleaf := pairedFrequencyLeaf_sum_le S V (fun s hs => (hS s hs).1) |
| calc |
| _ ≤ (8 * D ^ 4 * (1 + Real.log N) ^ 3) ^ (2 ^ n - 1) * |
| ((2 * (V : ℝ)) ^ 2) ^ (2 ^ n) := |
| mul_le_mul_of_nonneg_left (pow_le_pow_left₀ |
| (Finset.sum_nonneg fun x _ => by unfold pairedFrequencyLeaf; split_ifs <;> norm_num) |
| hleaf _) (pow_nonneg hC _) |
| _ = _ := by rw [← pow_mul] |
|
|
| end Ostmann |
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| end OAI |
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