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a72760f | 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 | 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
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