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caec230 | 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 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 | import OAI.MeasureTheory.DyadicAvoidance.GridGeometry
import OAI.MeasureTheory.DyadicAvoidance.GridMeasure
namespace OAI
universe u_ι
noncomputable section
namespace Problem310
open MeasureTheory Set
/-- Simultaneous stable centers for a finite list of earlier-grid/later-window pairs. -/
def gridStableCenters {ι : Type u_ι} (a b : ι → ℕ) : Set ℝ :=
(⋃ i, gridBadCenters ((2 : ℕ) ^ (b i + 2)) (2 * ((2 : ℝ)⁻¹) ^ (a i)))ᶜ
theorem measurableSet_gridStableCenters {ι : Type u_ι} [Countable ι] (a b : ι → ℕ) :
MeasurableSet (gridStableCenters a b) := by
apply MeasurableSet.compl
apply MeasurableSet.iUnion
intro i
exact measurableSet_gridBadCenters _ _
theorem gridStableCenters_add_one {ι : Type u_ι} (a b : ι → ℕ) (x : ℝ) :
x + 1 ∈ gridStableCenters a b ↔ x ∈ gridStableCenters a b := by
simp only [gridStableCenters, mem_compl_iff, mem_iUnion, not_exists]
apply forall_congr'
intro i
exact not_congr (gridBadCenters_add_one _ (by positivity) _ _)
/-- Stability at a grid also preserves every coarser grid at a later-window trial. -/
theorem gridStableCenters_preserves {ι : Type u_ι} {a b : ι → ℕ} {x t : ℝ}
(hx : x ∈ gridStableCenters a b) (i : ι) {c n : ℕ}
(hc : c ≤ b i) (hn : a i ≤ n) (ht : t ∈ Icc (1 : ℝ) 2) :
dyadicGridKey c (x + t * ((2 : ℝ)⁻¹) ^ n) = dyadicGridKey c x := by
have hxnot : x ∉ gridBadCenters ((2 : ℕ) ^ (b i + 2))
(2 * ((2 : ℝ)⁻¹) ^ (a i)) := by
intro hi
exact hx (mem_iUnion.mpr ⟨i, hi⟩)
have hkey := dyadicGridKey_eq_of_stable hn ht (fun k hk ↦ by
apply hxnot
apply (mem_gridBadCenters_iff _ _ _).mpr
exact ⟨k, by simpa using hk⟩)
rw [← dyadicGridKey_nested hc, ← dyadicGridKey_nested hc, hkey]
private theorem grid_window_width_le {a b : ℕ} (hgap : b + 3 ≤ a) :
2 * ((2 : ℝ)⁻¹) ^ a ≤ (((2 : ℕ) ^ (b + 2) : ℕ) : ℝ)⁻¹ := by
have h := pow_right_anti₀ (by norm_num : (0 : ℝ) ≤ 2⁻¹)
(by norm_num : (2 : ℝ)⁻¹ ≤ 1) hgap
change (2 : ℝ)⁻¹ ^ a ≤ (2 : ℝ)⁻¹ ^ (b + 3) at h
have hp : ((2 : ℝ)⁻¹) ^ (b + 3) = ((2 : ℝ)⁻¹) ^ (b + 2) * 2⁻¹ := by
rw [show b + 3 = (b + 2) + 1 by omega, pow_succ]
rw [hp] at h
have h2 : 2 * ((2 : ℝ)⁻¹) ^ a ≤ ((2 : ℝ)⁻¹) ^ (b + 2) := by
norm_num at h ⊢
linarith
simpa only [Nat.cast_pow, Nat.cast_ofNat, inv_pow] using h2
/-- A quantitative simultaneous stability estimate, with no tree assumptions. -/
theorem volume_gridStableCenters_compl_period_le {ι : Type u_ι} [Fintype ι]
(a b : ι → ℕ) (hgap : ∀ i, b i + 3 ≤ a i) :
volume ((gridStableCenters a b)ᶜ ∩ Icc (0 : ℝ) 1) ≤
ENNReal.ofReal (∑ i, (2 : ℝ) ^ (b i + 2) * (2 * ((2 : ℝ)⁻¹) ^ (a i))) := by
simp only [gridStableCenters, compl_compl]
simpa using volume_iUnion_gridBadCenters_period_le
(fun i ↦ (2 : ℕ) ^ (b i + 2)) (fun i ↦ 2 * ((2 : ℝ)⁻¹) ^ (a i))
(fun i ↦ by positivity) (fun i ↦ by positivity)
(fun i ↦ grid_window_width_le (hgap i))
/-- The density cost of a gap depends only on its length, not on the grid resolution. -/
theorem dyadic_grid_gap_budget (b g : ℕ) :
(2 : ℝ) ^ (b + 2) * (2 * ((2 : ℝ)⁻¹) ^ (b + g + 1)) =
4 * ((2 : ℝ)⁻¹) ^ g := by
have hcancel : (2 : ℝ) ^ b * ((2 : ℝ)⁻¹) ^ b = 1 := by
rw [inv_pow]
exact mul_inv_cancel₀ (by positivity)
calc
(2 : ℝ) ^ (b + 2) * (2 * ((2 : ℝ)⁻¹) ^ (b + g + 1))
= 4 * ((2 : ℝ) ^ b * ((2 : ℝ)⁻¹) ^ b) * ((2 : ℝ)⁻¹) ^ g := by
simp only [pow_add, pow_one]
norm_num
ring
_ = 4 * ((2 : ℝ)⁻¹) ^ g := by rw [hcancel]; ring
theorem volume_gridStableCenters_compl_gap_le {ι : Type u_ι} [Fintype ι]
(a b : ι → ℕ) (g : ℕ) (hg : 2 ≤ g)
(ha : ∀ i, a i = b i + g + 1) :
volume ((gridStableCenters a b)ᶜ ∩ Icc (0 : ℝ) 1) ≤
ENNReal.ofReal ((Fintype.card ι : ℝ) * (4 * ((2 : ℝ)⁻¹) ^ g)) := by
have h := volume_gridStableCenters_compl_period_le a b (fun i ↦ by rw [ha i]; omega)
have heq : (∑ i, (2 : ℝ) ^ (b i + 2) * (2 * ((2 : ℝ)⁻¹) ^ (a i))) =
(Fintype.card ι : ℝ) * (4 * ((2 : ℝ)⁻¹) ^ g) := by
simp only [ha, dyadic_grid_gap_budget, Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
rwa [heq] at h
/-- The same bound holds when windows are separated by at least the chosen gap.
This can be indexed by all ordered pairs of windows, not just predecessor pairs. -/
theorem volume_gridStableCenters_compl_gap_le_of_le {ι : Type u_ι} [Fintype ι]
(a b : ι → ℕ) (g : ℕ) (hg : 2 ≤ g)
(ha : ∀ i, b i + g + 1 ≤ a i) :
volume ((gridStableCenters a b)ᶜ ∩ Icc (0 : ℝ) 1) ≤
ENNReal.ofReal ((Fintype.card ι : ℝ) * (4 * ((2 : ℝ)⁻¹) ^ g)) := by
apply (volume_gridStableCenters_compl_period_le a b (fun i ↦ by have := ha i; omega)).trans
apply ENNReal.ofReal_le_ofReal
calc
(∑ i, (2 : ℝ) ^ (b i + 2) * (2 * ((2 : ℝ)⁻¹) ^ (a i)))
≤ ∑ i : ι, 4 * ((2 : ℝ)⁻¹) ^ g := by
apply Finset.sum_le_sum
intro i hi
have hp := pow_right_anti₀ (by norm_num : (0 : ℝ) ≤ 2⁻¹)
(by norm_num : (2 : ℝ)⁻¹ ≤ 1) (ha i)
calc
(2 : ℝ) ^ (b i + 2) * (2 * ((2 : ℝ)⁻¹) ^ (a i))
≤ (2 : ℝ) ^ (b i + 2) * (2 * ((2 : ℝ)⁻¹) ^ (b i + g + 1)) := by
exact mul_le_mul_of_nonneg_left
(mul_le_mul_of_nonneg_left hp (by norm_num)) (by positivity)
_ = 4 * ((2 : ℝ)⁻¹) ^ g := dyadic_grid_gap_budget (b i) g
_ = (Fintype.card ι : ℝ) * (4 * ((2 : ℝ)⁻¹) ^ g) := by
simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
/-- Any prescribed finite number of stability conditions has arbitrarily small
exceptional density once the unused gap is sufficiently long. -/
theorem exists_dyadic_gap_budget (K : ℕ) {p : ℝ} (hp : 0 < p) :
∃ g : ℕ, 2 ≤ g ∧ (K : ℝ) * (4 * ((2 : ℝ)⁻¹) ^ g) < p := by
obtain ⟨n, hn⟩ := exists_pow_lt_of_lt_one
(show 0 < p / (4 * (K + 1 : ℝ)) by positivity)
(show (2 : ℝ)⁻¹ < 1 by norm_num)
refine ⟨n + 2, by omega, ?_⟩
have hpow := pow_right_anti₀ (by norm_num : (0 : ℝ) ≤ 2⁻¹)
(by norm_num : (2 : ℝ)⁻¹ ≤ 1) (show n ≤ n + 2 by omega)
change (2 : ℝ)⁻¹ ^ (n + 2) ≤ (2 : ℝ)⁻¹ ^ n at hpow
have hden : 0 < 4 * (K + 1 : ℝ) := by positivity
have hlt := (lt_div_iff₀ hden).mp hn
have hk : (0 : ℝ) ≤ K := Nat.cast_nonneg K
have hnonneg : 0 ≤ ((2 : ℝ)⁻¹) ^ (n + 2) := by positivity
nlinarith
end Problem310
end
end OAI
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