| import OAI.MeasureTheory.DyadicAvoidance.GridStability |
| import OAI.MeasureTheory.DyadicAvoidance.GridSeparation |
|
|
| namespace OAI |
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|
| universe u_ι |
|
|
| noncomputable section |
| namespace Problem310 |
|
|
| / |
| 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 |
|
|
| / |
| 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 |
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|
| end |
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| end OAI |
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|
