| import Mathlib |
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| namespace OAI |
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| namespace Erdos970 |
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| section |
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| open Filter |
| open scoped Topology |
| namespace ErdosInverseBoxHeight |
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| theorem bin_inflation_le_power {B C xi : ℝ} (hB : 1 ≤ B) (hC : 0 ≤ C) |
| (hxi0 : 0 ≤ xi) (hxi1 : xi ≤ 1) : |
| (1+xi)^(C*Real.log B) ≤ B^C := by |
| have hB0 : 0 < B := by linarith |
| have hx0 : 0 < 1+xi := by linarith |
| have hlB : 0 ≤ Real.log B := Real.log_nonneg hB |
| have hlx : Real.log (1+xi) ≤ 1 := by |
| have h := Real.log_le_sub_one_of_pos hx0 |
| linarith |
| rw [Real.rpow_def_of_pos hx0,Real.rpow_def_of_pos hB0] |
| apply Real.exp_le_exp.mpr |
| nlinarith [mul_le_mul_of_nonneg_right hlx (mul_nonneg hC hlB)] |
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| theorem uniform_bin_inflation (C : ℝ) (hC : 0 ≤ C) : |
| ∀ᶠ z : ℝ in atTop, 1 < z ∧ ∀ B xi : ℝ, |
| 1 ≤ B → B ≤ Real.log z → 0 ≤ xi → xi ≤ 1 → |
| (1+xi)^(C*Real.log B) ≤ z^((1:ℝ)/10) := by |
| have hlog := (isLittleO_log_rpow_rpow_atTop C (by norm_num : (0:ℝ)<1/10)).bound zero_lt_one |
| filter_upwards [eventually_gt_atTop (1:ℝ),hlog] with z hz hpow |
| refine ⟨hz,?_⟩ |
| intro B xi hB hBz hxi0 hxi1 |
| have hz0 : 0 ≤ z := by linarith |
| have hL0 : 0 ≤ Real.log z := Real.log_nonneg hz.le |
| have hbound : (Real.log z)^C ≤ z^((1:ℝ)/10) := by |
| simpa only [Real.norm_eq_abs,abs_of_nonneg (Real.rpow_nonneg hL0 C), |
| abs_of_nonneg (Real.rpow_nonneg hz0 ((1:ℝ)/10)),one_mul] using hpow |
| exact (bin_inflation_le_power hB hC hxi0 hxi1).trans |
| ((Real.rpow_le_rpow (by linarith : 0 ≤ B) hBz hC).trans hbound) |
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|
| noncomputable def sourceZ (z : ℝ) : ℝ := Real.exp (Real.log z/Real.log (Real.log z)) |
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| theorem sourceZ_le_small_power : ∀ᶠ z : ℝ in atTop, |
| 1 < z ∧ 0 < sourceZ z ∧ sourceZ z ≤ z^((1:ℝ)/100) := by |
| have hlog := (Real.tendsto_log_atTop.comp Real.tendsto_log_atTop).eventually_ge_atTop 100 |
| filter_upwards [eventually_gt_atTop (1:ℝ),hlog] with z hz hL |
| dsimp only [Function.comp_apply] at hL |
| refine ⟨hz,Real.exp_pos _,?_⟩ |
| have hz0 : 0 < z := by linarith |
| have hlog0 : 0 ≤ Real.log z := (Real.log_pos hz).le |
| rw [sourceZ,Real.rpow_def_of_pos hz0] |
| apply Real.exp_le_exp.mpr |
| have h := div_le_div_of_nonneg_left hlog0 (by norm_num : (0:ℝ)<100) hL |
| convert! h using 1 |
| ring |
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| end ErdosInverseBoxHeight |
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| end |
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| end Erdos970 |
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| end OAI |
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