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namespace OAI
universe u_ι
noncomputable section
namespace Problem310
open MeasureTheory Set
/-- Centers whose positive displacement of length `δ` crosses a boundary
of the periodic grid with `q` cells. -/
def gridBadCenters (q : ℕ) (δ : ℝ) : Set ℝ :=
⋃ k : ℤ, Ico ((k : ℝ) / q - δ) ((k : ℝ) / q)
theorem mem_gridBadCenters_iff (q : ℕ) (δ x : ℝ) :
x ∈ gridBadCenters q δ ↔ ∃ k : ℤ, (k : ℝ) / q ∈ Ioc x (x + δ) := by
simp only [gridBadCenters, mem_iUnion, mem_Ico, mem_Ioc]
constructor
· rintro ⟨k, hk⟩
exact ⟨k, by linarith [hk.1], by linarith [hk.2]⟩
· rintro ⟨k, hk⟩
exact ⟨k, by linarith [hk.2], by linarith [hk.1]⟩
theorem measurableSet_gridBadCenters (q : ℕ) (δ : ℝ) :
MeasurableSet (gridBadCenters q δ) := by
apply MeasurableSet.iUnion
intro k
exact measurableSet_Ico
theorem gridBadCenters_add_one (q : ℕ) (hq : 0 < q) (δ x : ℝ) :
x + 1 ∈ gridBadCenters q δ ↔ x ∈ gridBadCenters q δ := by
rw [mem_gridBadCenters_iff, mem_gridBadCenters_iff]
have hq0 : (q : ℝ) ≠ 0 := by positivity
constructor
· rintro ⟨k, hk⟩
refine ⟨k - q, ?_⟩
have he : ((k - q : ℤ) : ℝ) / q = (k : ℝ) / q - 1 := by
push_cast
field_simp
rw [he]
exact ⟨by linarith [hk.1], by linarith [hk.2]⟩
· rintro ⟨k, hk⟩
refine ⟨k + q, ?_⟩
have he : ((k + q : ℤ) : ℝ) / q = (k : ℝ) / q + 1 := by
push_cast
field_simp
rw [he]
exact ⟨by linarith [hk.1], by linarith [hk.2]⟩
/-- Inside a half-open period the bad centers are covered by exactly `q`
intervals of length `δ`, provided `δ` is at most one cell width. -/
theorem gridBadCenters_period_subset (q : ℕ) (hq : 0 < q) {δ : ℝ}
(hδ : δ ≤ (q : ℝ)⁻¹) :
gridBadCenters q δ ∩ Ico (0 : ℝ) 1 ⊆
⋃ i : Fin q, Ico (((i.val + 1 : ℕ) : ℝ) / q - δ)
(((i.val + 1 : ℕ) : ℝ) / q) := by
intro x hx
obtain ⟨k, hk⟩ := (mem_gridBadCenters_iff q δ x).mp hx.1
have hqpos : (0 : ℝ) < q := by exact_mod_cast hq
have hkpos : 0 < k := by
have : (0 : ℝ) < (k : ℝ) / q := lt_of_le_of_lt hx.2.1 hk.1
have := (div_pos_iff.mp this)
exact_mod_cast (by rcases this with h | h; exact h.1; linarith [h.2] : (0 : ℝ) < k)
have hklt : k < (q : ℤ) + 1 := by
have hstep : (k : ℝ) / q < 1 + (q : ℝ)⁻¹ := by linarith [hx.2.2, hk.2]
have hmul := (div_lt_iff₀ hqpos).mp hstep
have hi : (q : ℝ)⁻¹ * q = 1 := inv_mul_cancel₀ hqpos.ne'
have : (k : ℝ) < q + 1 := by nlinarith
exact_mod_cast this
have hkq : k.toNat ≤ q := by omega
have hkn : 0 < k.toNat := by omega
let i : Fin q := ⟨k.toNat - 1, by omega⟩
apply mem_iUnion.mpr
refine ⟨i, ?_⟩
have hki : ((i.val + 1 : ℕ) : ℝ) = (k : ℝ) := by
have : i.val + 1 = k.toNat := by dsimp [i]; omega
rw [this]
exact_mod_cast (Int.toNat_of_nonneg hkpos.le)
rw [hki]
exact ⟨by linarith [hk.2], hk.1⟩
theorem volume_gridBadCenters_period_le (q : ℕ) (hq : 0 < q) {δ : ℝ}
(_hδ0 : 0 ≤ δ) (hδ : δ ≤ (q : ℝ)⁻¹) :
volume (gridBadCenters q δ ∩ Icc (0 : ℝ) 1) ≤ ENNReal.ofReal ((q : ℝ) * δ) := by
have hae : (gridBadCenters q δ ∩ Icc (0 : ℝ) 1 : Set ℝ) =ᵐ[volume]
(gridBadCenters q δ ∩ Ico (0 : ℝ) 1 : Set ℝ) := by
exact Filter.EventuallyEqSet.inter Filter.EventuallyEq.rfl
(Ico_ae_eq_Icc (μ := volume) (a := (0 : ℝ)) (b := 1)).symm
rw [measure_congr hae]
calc
volume (gridBadCenters q δ ∩ Ico (0 : ℝ) 1)
≤ volume (⋃ i : Fin q, Ico (((i.val + 1 : ℕ) : ℝ) / q - δ)
(((i.val + 1 : ℕ) : ℝ) / q)) :=
measure_mono (gridBadCenters_period_subset q hq hδ)
_ ≤ ∑ i : Fin q, volume (Ico (((i.val + 1 : ℕ) : ℝ) / q - δ)
(((i.val + 1 : ℕ) : ℝ) / q)) := measure_iUnion_fintype_le _ _
_ = ENNReal.ofReal ((q : ℝ) * δ) := by
simp only [Real.volume_Ico, sub_sub_cancel]
simp [ENNReal.ofReal_mul]
/-- The exceptional density for any finite family of stability requirements
is bounded by the sum of its grid-width budgets. -/
theorem volume_iUnion_gridBadCenters_period_le {ι : Type u_ι} [Fintype ι]
(q : ι → ℕ) (δ : ι → ℝ) (hq : ∀ i, 0 < q i)
(hδ0 : ∀ i, 0 ≤ δ i) (hδ : ∀ i, δ i ≤ (q i : ℝ)⁻¹) :
volume ((⋃ i, gridBadCenters (q i) (δ i)) ∩ Icc (0 : ℝ) 1) ≤
ENNReal.ofReal (∑ i, (q i : ℝ) * δ i) := by
classical
rw [iUnion_inter]
calc
volume (⋃ i, gridBadCenters (q i) (δ i) ∩ Icc (0 : ℝ) 1)
≤ ∑ i, volume (gridBadCenters (q i) (δ i) ∩ Icc (0 : ℝ) 1) :=
measure_iUnion_fintype_le _ _
_ ≤ ∑ i, ENNReal.ofReal ((q i : ℝ) * δ i) := by
apply Finset.sum_le_sum
intro i hi
exact volume_gridBadCenters_period_le (q i) (hq i) (hδ0 i) (hδ i)
_ = ENNReal.ofReal (∑ i, (q i : ℝ) * δ i) := by
symm
apply ENNReal.ofReal_sum_of_nonneg
intro i hi
exact mul_nonneg (Nat.cast_nonneg _) (hδ0 i)
end Problem310
end
end OAI
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