| import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics |
| import Mathlib.Analysis.SpecificLimits.Normed |
| import Mathlib.Tactic |
|
|
| namespace OAI |
|
|
| open Filter |
| open scoped Topology BigOperators |
|
|
| namespace Balanced.LocalAsymptotics |
|
|
| theorem summable_polynomial_mul_pow_div_factorial (a : ℕ) (C : ℝ) (hC : 0 ≤ C) : |
| Summable (fun b : ℕ => (b : ℝ) ^ a * C ^ b / (b.factorial : ℝ)) := by |
| apply (Real.summable_pow_div_factorial (2 * C)).of_norm_bounded_eventually_nat |
| have h := (tendsto_pow_const_div_const_pow_of_one_lt a (by norm_num : (1 : ℝ) < 2)).eventually |
| (gt_mem_nhds (by norm_num : (0 : ℝ) < 1)) |
| filter_upwards [h] with b hb |
| have hb' : (b : ℝ) ^ a ≤ (2 : ℝ) ^ b := by |
| exact le_of_lt ((div_lt_one (by positivity : (0 : ℝ) < 2 ^ b)).mp hb) |
| rw [Real.norm_of_nonneg (by positivity), mul_pow] |
| exact div_le_div_of_nonneg_right (mul_le_mul_of_nonneg_right hb' (pow_nonneg hC b)) |
| (by positivity) |
|
|
| theorem tendsto_log_pow_div (a : ℕ) : |
| Tendsto (fun x : ℝ => (Real.log x) ^ a / x) atTop (𝓝 0) := by |
| have h := isLittleO_log_rpow_rpow_atTop (a : ℝ) (by norm_num : (0 : ℝ) < 1) |
| simpa only [Real.rpow_natCast, Real.rpow_one] using h.tendsto_div_nhds_zero |
|
|
| theorem tendsto_ceil_log_pow_div (a : ℕ) : |
| Tendsto (fun x : ℝ => (⌈Real.log x⌉₊ : ℝ) ^ a / x) atTop (𝓝 0) := by |
| have hlog : ∀ᶠ x : ℝ in atTop, 1 ≤ Real.log x := |
| Real.tendsto_log_atTop.eventually_ge_atTop 1 |
| have hx : ∀ᶠ x : ℝ in atTop, 0 ≤ x := eventually_ge_atTop 0 |
| apply squeeze_zero' (hx.mono fun x hx => by positivity) |
| (g := fun x : ℝ => (2 : ℝ) ^ a * ((Real.log x) ^ a / x)) |
| · filter_upwards [hlog, hx] with x hlog hx |
| have hceil : (⌈Real.log x⌉₊ : ℝ) ≤ 2 * Real.log x := by |
| have h := Nat.ceil_lt_add_one (le_trans (by norm_num : (0 : ℝ) ≤ 1) hlog) |
| linarith |
| calc |
| (⌈Real.log x⌉₊ : ℝ) ^ a / x ≤ (2 * Real.log x) ^ a / x := by gcongr |
| _ = (2 : ℝ) ^ a * ((Real.log x) ^ a / x) := by rw [mul_pow]; ring |
| · simpa using (tendsto_log_pow_div a).const_mul ((2 : ℝ) ^ a) |
|
|
| theorem tendsto_ceil_log_cube_div_nat : |
| Tendsto (fun q : ℕ => (⌈Real.log (q : ℝ)⌉₊ : ℝ) ^ 3 / (q : ℝ)) |
| atTop (𝓝 0) := |
| (tendsto_ceil_log_pow_div 3).comp tendsto_natCast_atTop_atTop |
|
|
| theorem large_pencil_tail_le_inv (C x : ℝ) (hC : 0 ≤ C) (hx : 0 < x) |
| (hlog : 1 ≤ Real.log x) |
| (hlarge : C * Real.exp 3 ≤ (⌈Real.log x⌉₊ : ℝ)) : |
| x ^ 2 * (C / (⌈Real.log x⌉₊ : ℝ)) ^ ⌈Real.log x⌉₊ ≤ x⁻¹ := by |
| have hceil := Nat.le_ceil (Real.log x) |
| have hB : (0 : ℝ) < (⌈Real.log x⌉₊ : ℝ) := by linarith |
| have hratio : C / (⌈Real.log x⌉₊ : ℝ) ≤ Real.exp (-3) := by |
| apply (div_le_iff₀ hB).2 |
| calc |
| C = (C * Real.exp 3) * Real.exp (-3) := by |
| rw [mul_assoc, ← Real.exp_add] |
| norm_num |
| _ ≤ (⌈Real.log x⌉₊ : ℝ) * Real.exp (-3) := |
| mul_le_mul_of_nonneg_right hlarge (Real.exp_pos _).le |
| _ = Real.exp (-3) * (⌈Real.log x⌉₊ : ℝ) := mul_comm _ _ |
| have hexp : Real.exp (-3 * Real.log x) = (x ^ 3)⁻¹ := by |
| rw [show -3 * Real.log x = -(3 * Real.log x) by ring, Real.exp_neg] |
| congr 1 |
| simpa [Real.exp_log hx] using Real.exp_nat_mul (Real.log x) 3 |
| calc |
| x ^ 2 * (C / (⌈Real.log x⌉₊ : ℝ)) ^ ⌈Real.log x⌉₊ |
| ≤ x ^ 2 * Real.exp (-3) ^ ⌈Real.log x⌉₊ := by gcongr |
| _ = x ^ 2 * Real.exp (-3 * (⌈Real.log x⌉₊ : ℝ)) := by |
| rw [mul_comm (-3), Real.exp_nat_mul] |
| _ ≤ x ^ 2 * Real.exp (-3 * Real.log x) := by |
| apply mul_le_mul_of_nonneg_left _ (sq_nonneg x) |
| apply Real.exp_le_exp.mpr |
| linarith |
| _ = x⁻¹ := by rw [hexp]; field_simp |
|
|
| theorem tendsto_large_pencil_tail (C : ℝ) (hC : 0 ≤ C) : |
| Tendsto (fun x : ℝ => x ^ 2 * |
| (C / (⌈Real.log x⌉₊ : ℝ)) ^ ⌈Real.log x⌉₊) atTop (𝓝 0) := by |
| apply squeeze_zero' (Eventually.of_forall fun x => by positivity) |
| (g := fun x : ℝ => x⁻¹) |
| · filter_upwards [eventually_gt_atTop (0 : ℝ), |
| Real.tendsto_log_atTop.eventually_ge_atTop 1, |
| Real.tendsto_log_atTop.eventually_ge_atTop (C * Real.exp 3)] with x hx hlog hlarge |
| exact large_pencil_tail_le_inv C x hC hx hlog |
| (hlarge.trans (Nat.le_ceil (Real.log x))) |
| · exact tendsto_inv_atTop_zero |
|
|
| theorem tendsto_large_pencil_tail_nat (C : ℝ) (hC : 0 ≤ C) : |
| Tendsto (fun q : ℕ => (q : ℝ) ^ 2 * |
| (C / (⌈Real.log (q : ℝ)⌉₊ : ℝ)) ^ ⌈Real.log (q : ℝ)⌉₊) |
| atTop (𝓝 0) := |
| (tendsto_large_pencil_tail C hC).comp tendsto_natCast_atTop_atTop |
|
|
| theorem sum_polynomial_mul_pow_div_factorial_le (a : ℕ) (C : ℝ) (hC : 0 ≤ C) |
| (s : Finset ℕ) : |
| (∑ b ∈ s, (b : ℝ) ^ a * C ^ b / (b.factorial : ℝ)) ≤ |
| ∑' b : ℕ, (b : ℝ) ^ a * C ^ b / (b.factorial : ℝ) := by |
| exact (summable_polynomial_mul_pow_div_factorial a C hC).sum_le_tsum s |
| (fun b _ => by positivity) |
|
|
| theorem tendsto_weighted_log_bound (a : ℕ) (C : ℝ) (hC : 0 ≤ C) |
| (s : ℕ → Finset ℕ) : |
| Tendsto (fun q : ℕ => |
| (∑ b ∈ s q, (b : ℝ) ^ a * C ^ b / (b.factorial : ℝ)) * |
| ((⌈Real.log (q : ℝ)⌉₊ : ℝ) ^ 3 / (q : ℝ))) atTop (𝓝 0) := by |
| apply squeeze_zero (fun q => by positivity) |
| (g := fun q : ℕ => (∑' b : ℕ, (b : ℝ) ^ a * C ^ b / (b.factorial : ℝ)) * |
| ((⌈Real.log (q : ℝ)⌉₊ : ℝ) ^ 3 / (q : ℝ))) |
| · intro q |
| apply mul_le_mul_of_nonneg_right |
| (sum_polynomial_mul_pow_div_factorial_le a C hC (s q)) |
| positivity |
| · simpa using tendsto_ceil_log_cube_div_nat.const_mul |
| (∑' b : ℕ, (b : ℝ) ^ a * C ^ b / (b.factorial : ℝ)) |
|
|
| theorem tendsto_plane_weighted_pencil_sum (a : ℕ) (C : ℝ) (hC : 0 ≤ C) |
| (s : ℕ → Finset ℕ) : |
| Tendsto (fun q : ℕ => (q : ℝ) ^ 2 * |
| ∑ b ∈ s q, ((b : ℝ) ^ a * C ^ b / (b.factorial : ℝ)) * |
| ((⌈Real.log (q : ℝ)⌉₊ : ℝ) ^ 3 / (q : ℝ) ^ 3)) atTop (𝓝 0) := by |
| apply (tendsto_weighted_log_bound a C hC s).congr' |
| filter_upwards [eventually_gt_atTop (0 : ℕ)] with q hq |
| have hq' : (q : ℝ) ≠ 0 := by exact_mod_cast (Nat.ne_of_gt hq) |
| rw [← Finset.sum_mul] |
| field_simp |
|
|
| theorem pencil_common_factor_le (q K : ℝ) (b κ : ℕ) |
| (hq : 1 ≤ q) (hK : 0 ≤ K) (hκ : κ ≤ 5 * b) : |
| (32 : ℝ) ^ b * (q + 1) ^ (5 * b - κ) * (K / q) ^ b / q ^ (4 * b) ≤ |
| (1024 * K) ^ b / q ^ κ := by |
| have hqpos : 0 < q := lt_of_lt_of_le zero_lt_one hq |
| have hqne : q ≠ 0 := ne_of_gt hqpos |
| have hp : (q + 1) ^ (5 * b - κ) ≤ (2 : ℝ) ^ (5 * b) * q ^ (5 * b - κ) := by |
| calc |
| (q + 1) ^ (5 * b - κ) ≤ (2 * q) ^ (5 * b - κ) := by gcongr; linarith |
| _ = (2 : ℝ) ^ (5 * b - κ) * q ^ (5 * b - κ) := mul_pow _ _ _ |
| _ ≤ (2 : ℝ) ^ (5 * b) * q ^ (5 * b - κ) := |
| mul_le_mul_of_nonneg_right |
| (pow_le_pow_right₀ (by norm_num : (1 : ℝ) ≤ 2) (Nat.sub_le _ _)) |
| (pow_nonneg hqpos.le _) |
| have hqp : q ^ (5 * b) = q ^ b * q ^ (4 * b) := by |
| rw [← pow_add] |
| congr 1 |
| omega |
| have hc : (32 : ℝ) ^ b * (2 : ℝ) ^ (5 * b) * K ^ b = (1024 * K) ^ b := by |
| rw [pow_mul, ← mul_pow, ← mul_pow] |
| norm_num |
| calc |
| (32 : ℝ) ^ b * (q + 1) ^ (5 * b - κ) * (K / q) ^ b / q ^ (4 * b) |
| ≤ (32 : ℝ) ^ b * ((2 : ℝ) ^ (5 * b) * q ^ (5 * b - κ)) * |
| (K / q) ^ b / q ^ (4 * b) := by gcongr |
| _ = (1024 * K) ^ b / q ^ κ := by |
| rw [div_pow, pow_sub₀ q hqne hκ, hqp] |
| field_simp |
| nlinarith [hc] |
|
|
| end Balanced.LocalAsymptotics |
|
|
| end OAI |
|
|
