| import OAI.NumberTheory.PiExponent.Polynomials.SimplexFloor |
| import OAI.NumberTheory.PiExponent.Polynomials.SimplexLog |
|
|
| namespace OAI |
|
|
| open Filter Topology |
|
|
| namespace PiExponent |
|
|
| noncomputable def translationTermCount (m : ℕ) (v H : ℝ) : ℕ := |
| (⌊H⌋₊ + 1) ^ (2 * m) * (⌊H / v⌋₊ + 1) |
|
|
| theorem tendsto_natFloor_mul_add_one_div {a : ℝ} (ha : 0 < a) : |
| Tendsto (fun H : ℝ => ((⌊a * H⌋₊ + 1 : ℕ) : ℝ) / H) atTop (𝓝 a) := by |
| have h := (tendsto_natFloor_affine_div ha 0).add (tendsto_id.const_div_atTop (1 : ℝ)) |
| simpa only [add_zero, id_eq, Nat.cast_add, Nat.cast_one, add_div] using h |
|
|
| theorem tendsto_translationTermCount_normalized (m : ℕ) {v : ℝ} (hv : 0 < v) : |
| Tendsto (fun H : ℝ => (translationTermCount m v H : ℝ) / H ^ (2 * m + 1)) |
| atTop (𝓝 (1 / v)) := by |
| have h₁ : Tendsto (fun H : ℝ => ((⌊H⌋₊ + 1 : ℕ) : ℝ) / H) atTop (𝓝 1) := by |
| simpa only [one_mul] using tendsto_natFloor_mul_add_one_div (show (0 : ℝ) < 1 by norm_num) |
| have h₂ : Tendsto (fun H : ℝ => ((⌊H / v⌋₊ + 1 : ℕ) : ℝ) / H) |
| atTop (𝓝 (1 / v)) := by |
| simpa only [one_div, div_eq_mul_inv, mul_comm, mul_one, one_mul] using |
| tendsto_natFloor_mul_add_one_div (div_pos zero_lt_one hv) |
| have h := (h₁.pow (2 * m)).mul h₂ |
| simp only [one_pow, one_mul] at h |
| convert h using 1 |
| ext H |
| simp only [translationTermCount, Nat.cast_mul, Nat.cast_pow, div_pow, div_mul_div_comm, |
| pow_succ] |
|
|
| theorem tendsto_log_translationTermCount_div (m : ℕ) {v : ℝ} (hv : 0 < v) : |
| Tendsto (fun H : ℝ => Real.log (translationTermCount m v H : ℝ) / H) |
| atTop (𝓝 0) := |
| tendsto_log_div_of_normalized_pow (tendsto_translationTermCount_normalized m hv) |
| (div_pos zero_lt_one hv) |
|
|
| theorem tendsto_log_translationTermCount_div_nat (m : ℕ) {v : ℝ} (hv : 0 < v) : |
| Tendsto (fun H : ℕ => Real.log (translationTermCount m v (H : ℝ) : ℝ) / (H : ℝ)) |
| atTop (𝓝 0) := |
| (tendsto_log_translationTermCount_div m hv).comp tendsto_natCast_atTop_atTop |
|
|
| end PiExponent |
|
|
| end OAI |
|
|
