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d4317d7 | 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 | import OAI.MeasureTheory.Falconer.Graphs.WeakGraphObstruction
import OAI.MeasureTheory.Falconer.Estimates.ScalarWeakRefinement
namespace OAI
/-! Numerical absorption for passing from dense original graphs to the
common refined-graph obstruction. -/
namespace Falconer
theorem dense_graph_parameters {θ : ℝ} (hθ : 0 < θ) :
∃ γ : ℝ, ∃ N0 : ℕ, 0 < γ ∧ 0 < N0 ∧ ∀ N : ℕ, N0 ≤ N →
let S := Real.exp (γ*(N:ℝ)*Real.log 2)
let R := Real.exp (θ*(N:ℝ)*Real.log 2)
1 ≤ S ∧ S ≤ R ∧ 67108864*S^13 ≤ R ∧ R⁻¹ ≤ (S⁻¹)^2/64 := by
obtain ⟨N0,hN0,hbounds⟩ := scalar_fixed_losses hθ
(by norm_num : (0:ℝ) < 67108864) (by norm_num : (0:ℝ) < 64)
let γ := θ/100
have hγ : 0 < γ := by dsimp [γ]; positivity
refine ⟨γ,N0,hγ,hN0,?_⟩
intro N hN
dsimp only
let S := Real.exp (γ*(N:ℝ)*Real.log 2)
let R := Real.exp (θ*(N:ℝ)*Real.log 2)
have hSpos : 0 < S := Real.exp_pos _
have hRpos : 0 < R := Real.exp_pos _
have hlog : 0 ≤ (N:ℝ)*Real.log 2 := by positivity
have hS : 1 ≤ S := Real.one_le_exp_iff.mpr (by positivity)
have hSR : S ≤ R := by
apply Real.exp_le_exp.mpr
dsimp [γ]
nlinarith
have hp (n : ℕ) (hn : (n:ℝ) ≤ 25) : S^n ≤ Real.exp ((θ/4)*(N:ℝ)*Real.log 2) := by
rw [show S = Real.exp (γ*(N:ℝ)*Real.log 2) from rfl,← Real.exp_nat_mul]
apply Real.exp_le_exp.mpr
dsimp [γ]
have hh := mul_le_mul_of_nonneg_right hn (mul_nonneg hθ.le hlog)
nlinarith
have h13 : 67108864*S^13 ≤ R :=
(mul_le_mul_of_nonneg_left (hp 13 (by norm_num)) (by norm_num)).trans (hbounds N hN).1
have h2 : 64*S^2 ≤ R :=
(mul_le_mul_of_nonneg_left (hp 2 (by norm_num)) (by norm_num)).trans (hbounds N hN).2
refine ⟨hS,hSR,h13,?_⟩
have hi := one_div_le_one_div_of_le (by positivity : 0 < 64*S^2) h2
have he : (S⁻¹)^2/64 = 1/(64*S^2) := by field_simp
change R⁻¹ ≤ (S⁻¹)^2/64
rw [he]
simpa only [one_div] using hi
end Falconer
end OAI
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