File size: 2,163 Bytes
f89a930 | 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 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 | import Mathlib
import OAI.NumberTheory.Jacobsthal.Estimates.UniformCollisionError
namespace OAI
namespace Erdos970
section
namespace ErdosLineCollision
theorem normalized_collision_bound (I K C n R N L : ℝ)
(hK0 : 0 ≤ K) (hC : 0 ≤ C) (hn : 0 < n) (hN : 0 < N) (hL : 0 < L)
(hcollision : I^2 ≤ K*(C*N^2+I)) (hthin : K ≤ n*R/L^6) (hI : I ≤ n*N) :
(I/(n*N))^2 ≤ C*R/(n*L^6)+R/(N*L^6) := by
have hup : I^2 ≤ (n*R/L^6)*(C*N^2+n*N) := by
calc
_ ≤ K*(C*N^2+I) := hcollision
_ ≤ K*(C*N^2+n*N) := mul_le_mul_of_nonneg_left (add_le_add le_rfl hI) hK0
_ ≤ _ := mul_le_mul_of_nonneg_right hthin (by positivity)
rw [div_pow]
calc
_ ≤ ((n*R/L^6)*(C*N^2+n*N))/(n*N)^2 :=
div_le_div_of_nonneg_right hup (sq_nonneg _)
_ = _ := by field_simp
end ErdosLineCollision
end
section
open Filter
open scoped Topology
namespace ErdosSourceCollision
open ErdosLineCollision
theorem uniform_normalized_collision (C eta eps : ℝ) (hC : 0 ≤ C) (heta : 0 < eta) (heps : 0 < eps) :
∀ᶠ z : ℝ in atTop, 1 < z ∧ ∀ inc K R N n : ℝ,
1 < R → R ≤ z^((1 : ℝ)/100) → z^((93 : ℝ)/100) ≤ N →
eta*R/(4*Real.log R) ≤ n → 0 ≤ K →
inc^2 ≤ K*(C*N^2+inc) → K ≤ n*R/(Real.log z)^6 → inc ≤ n*N →
inc ≤ eps*n*N := by
filter_upwards [uniform_collision_error C eta eps hC heta heps] with z hz
refine ⟨hz.1,?_⟩
intro inc K R N n hR hRz hN hcount hK hcoll hthin hI
have hz0 : 0 < z := by linarith [hz.1]
have hR0 : 0 < R := by linarith
have hlogR : 0 < Real.log R := Real.log_pos hR
have hn : 0 < n := (div_pos (mul_pos heta hR0) (mul_pos (by norm_num) hlogR)).trans_le hcount
have hNp : 0 < N := (Real.rpow_pos_of_pos hz0 _).trans_le hN
have hL : 0 < Real.log z := by linarith [hz.2.1]
have hsq := (normalized_collision_bound inc K C n R N (Real.log z) hK hC hn hNp hL hcoll hthin hI).trans
(hz.2.2 R N n hR hRz hN hcount)
have hratio : inc/(n*N) ≤ eps := by nlinarith [sq_nonneg (inc/(n*N)-eps)]
have hh := (div_le_iff₀ (mul_pos hn hNp)).mp hratio
nlinarith
end ErdosSourceCollision
end
end Erdos970
end OAI
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