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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 | import OAI.MeasureTheory.DyadicAvoidance.GridStability
import OAI.MeasureTheory.DyadicAvoidance.GridSeparation
namespace OAI
universe u_ι
noncomputable section
namespace Problem310
/-- Bridge from quantitative stability to the canonical finite selector addresses. -/
theorem gridStableCenters_preserves_fin {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 ∈ Set.Icc (1 : ℝ) 2) :
GridSeparation.dyadicKey c (x + t * ((2 : ℝ)⁻¹) ^ n) =
GridSeparation.dyadicKey c x := by
apply (GridSeparation.dyadicKey_eq_iff c _ _).mpr
exact gridStableCenters_preserves hx i hc hn ht
/-- Choose the gap using only the finite number of grids, before their resolutions
are assigned. Every subsequent assignment has a periodic measurable stable set
with the same prescribed exceptional-density budget. -/
theorem exists_uniform_grid_stability {ι : Type u_ι} [Fintype ι]
(p : ℝ) (hp : 0 < p) :
∃ g : ℕ, 2 ≤ g ∧ ∀ b : ι → ℕ, ∃ G : Set ℝ,
MeasurableSet G ∧ (∀ x : ℝ, x + 1 ∈ G ↔ x ∈ G) ∧
MeasureTheory.volume (Gᶜ ∩ Set.Icc (0 : ℝ) 1) ≤ ENNReal.ofReal p ∧
∀ x ∈ G, ∀ i : ι, ∀ n : ℕ, b i + g + 1 ≤ n →
∀ t ∈ Set.Icc (1 : ℝ) 2,
GridSeparation.dyadicKey (b i) (x + t * ((2 : ℝ)⁻¹) ^ n) =
GridSeparation.dyadicKey (b i) x := by
obtain ⟨g, hg, hbudget⟩ := exists_dyadic_gap_budget (Fintype.card ι) hp
refine ⟨g, hg, fun b ↦ ?_⟩
let a : ι → ℕ := fun i ↦ b i + g + 1
refine ⟨gridStableCenters a b, measurableSet_gridStableCenters a b,
gridStableCenters_add_one a b, ?_, ?_⟩
· exact (volume_gridStableCenters_compl_gap_le a b g hg (fun i ↦ rfl)).trans
(ENNReal.ofReal_le_ofReal hbudget.le)
· intro x hx i n hn t ht
exact gridStableCenters_preserves_fin hx i le_rfl hn ht
end Problem310
end
end OAI
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