| import OAI.NumberTheory.TwoPoint.Fourier.MinorArcRationalSpacing |
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| namespace OAI |
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| /-! Elementary interval packing for separated phase representatives. -/ |
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| namespace TwoPointCorrelations |
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| open Finset |
| open scoped Classical |
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| lemma minor_arc_interval_packing {ι : Type*} (S : Finset ι) (x : ι → ℝ) |
| (δ b : ℝ) (hsep : ∀ i ∈ S, ∀ j ∈ S, i ≠ j → δ ≤ |x i - x j|) : |
| (S.filter (fun i => b ≤ x i ∧ x i < b + δ)).card ≤ 1 := by |
| apply card_le_one.mpr |
| intro i hi j hj |
| obtain ⟨hiS, hix⟩ := mem_filter.mp hi |
| obtain ⟨hjS, hjx⟩ := mem_filter.mp hj |
| by_contra hij |
| have hh := hsep i hiS j hjS hij |
| have habs : |x i - x j| < δ := abs_lt.mpr ⟨by linarith, by linarith⟩ |
| linarith |
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| lemma minor_arc_absolute_band_packing {ι : Type*} (S : Finset ι) (x : ι → ℝ) |
| (δ b : ℝ) |
| (hsep : ∀ i ∈ S, ∀ j ∈ S, i ≠ j → δ ≤ |x i - x j|) : |
| (S.filter (fun i => b ≤ |x i| ∧ |x i| < b + δ)).card ≤ 2 := by |
| let P := S.filter (fun i => b ≤ x i ∧ x i < b + δ) |
| let Q := S.filter (fun i => b ≤ -x i ∧ -x i < b + δ) |
| have hP : P.card ≤ 1 := minor_arc_interval_packing S x δ b hsep |
| have hQ : Q.card ≤ 1 := by |
| apply minor_arc_interval_packing S (fun i => -x i) δ b |
| intro i hi j hj hij |
| simpa only [neg_sub_neg, abs_sub_comm] using hsep i hi j hj hij |
| have hsub : S.filter (fun i => b ≤ |x i| ∧ |x i| < b + δ) ⊆ P ∪ Q := by |
| intro i hi |
| obtain ⟨hiS, hix⟩ := mem_filter.mp hi |
| by_cases hx : 0 ≤ x i |
| · apply mem_union_left |
| exact mem_filter.mpr ⟨hiS, by simpa only [abs_of_nonneg hx] using hix⟩ |
| · apply mem_union_right |
| exact mem_filter.mpr ⟨hiS, by simpa only [abs_of_neg (lt_of_not_ge hx)] using hix⟩ |
| have hu := (card_le_card hsub).trans (card_union_le P Q) |
| omega |
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| end TwoPointCorrelations |
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| end OAI |
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