| import OAI.MeasureTheory.DyadicAvoidance.GridGeometry |
| import OAI.MeasureTheory.DyadicAvoidance.GridMeasure |
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| namespace OAI |
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| universe u_ι |
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| noncomputable section |
| namespace Problem310 |
| open MeasureTheory Set |
|
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| / |
| 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 |
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|
| 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) _ _) |
|
|
| / |
| 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 |
|
|
| / |
| 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)) |
|
|
| / |
| 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 |
|
|
| / |
| 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] |
|
|
| / |
| 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 |
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| end Problem310 |
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| end |
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| end OAI |
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