| import OAI.GroupTheory.Thompson.AnalyticSinc |
| import Mathlib.Analysis.Normed.Group.Basic |
|
|
| namespace OAI |
|
|
| /-! |
| # Finite-region uniform separation |
|
|
| The generic results below separate the compactness and scalar estimates from |
| the concrete curve calculations. The actual analytic curve must supply every |
| listed identity and inequality before this result is used in the tube theorem. |
| -/ |
|
|
| noncomputable section |
|
|
| universe uH |
|
|
| namespace ThompsonNonamenability.Analytic |
|
|
| open Set |
|
|
| theorem exists_uniform_sep_on_square |
| {H : Type uH} [NormedAddCommGroup H] |
| (γ : ℝ → H) (hcont : Continuous γ) (hinj : Function.Injective γ) |
| (d : ℝ) (hd : 0 < d) : |
| ∃ μ : ℝ, 0 < μ ∧ |
| ∀ s ∈ Icc (-4 : ℝ) 5, |
| ∀ t ∈ Icc (-4 : ℝ) 5, |
| d ≤ t - s → μ ≤ ‖γ t - γ s‖ := by |
| let C : Set (ℝ × ℝ) := |
| (Icc (-4 : ℝ) 5 ×ˢ Icc (-4 : ℝ) 5) ∩ {p | d ≤ p.2 - p.1} |
| have hclosed : IsClosed {p : ℝ × ℝ | d ≤ p.2 - p.1} := |
| isClosed_le continuous_const (continuous_snd.sub continuous_fst) |
| have hcompact : IsCompact C := |
| ((isCompact_Icc : IsCompact (Icc (-4 : ℝ) 5)).prod isCompact_Icc).inter_right |
| hclosed |
| have hcontinuous : Continuous (fun p : ℝ × ℝ => ‖γ p.2 - γ p.1‖) := |
| ((hcont.comp continuous_snd).sub (hcont.comp continuous_fst)).norm |
| have hpositive : ∀ p ∈ C, 0 < ‖γ p.2 - γ p.1‖ := by |
| intro p hp |
| apply norm_pos_iff.mpr |
| apply sub_ne_zero.mpr |
| intro heq |
| have hparam : p.2 = p.1 := hinj heq |
| have hgap : d ≤ p.2 - p.1 := hp.2 |
| rw [hparam, sub_self] at hgap |
| exact (not_le_of_gt hd) hgap |
| obtain ⟨μ, hμ, hbound⟩ := hcompact.exists_forall_le' |
| hcontinuous.continuousOn hpositive |
| refine ⟨μ, hμ, ?_⟩ |
| intro s hs t ht hgap |
| exact hbound (s, t) ⟨⟨hs, ht⟩, hgap⟩ |
|
|
| theorem uniform_separation_of_five_regions |
| {H : Type uH} [NormedAddCommGroup H] |
| (γ : ℝ → H) (hcont : Continuous γ) (hinj : Function.Injective γ) |
| (hline : ∀ s t : ℝ, s ≤ t → t ≤ 1 → ‖γ t - γ s‖ = (t - s) / 8) |
| (hnegative : ∀ s t : ℝ, s ≤ -4 → 1 < t → 3 / 16 ≤ ‖γ t - γ s‖) |
| (htail : ∀ s t : ℝ, 3 ≤ s → s ≤ t → |
| ‖γ t - γ s‖ ^ 2 = 2 * (5 / 16 : ℝ) ^ 2 * (1 - sinc (t - s))) |
| (hmixed : ∀ s t : ℝ, s ≤ 3 → 5 ≤ t → |
| 3 * (5 / 16 : ℝ) ^ 2 / 4 ≤ ‖γ t - γ s‖ ^ 2) |
| (d : ℝ) (hd : 0 < d) : |
| ∃ η : ℝ, 0 < η ∧ ∀ s t : ℝ, d ≤ |t - s| → η ≤ ‖γ t - γ s‖ := by |
| obtain ⟨ε, hε, hεhalf, hsinc⟩ := exists_uniform_sinc_gap d hd |
| obtain ⟨μ, hμ, hcompact⟩ := exists_uniform_sep_on_square γ hcont hinj d hd |
| let η : ℝ := min (d / 8) (min (3 / 16) (min ((5 / 16) * ε) (min (5 / 32) μ))) |
| have hηpos : 0 < η := by |
| dsimp [η] |
| exact lt_min (div_pos hd (by norm_num)) |
| (lt_min (by norm_num) (lt_min (mul_pos (by norm_num) hε) |
| (lt_min (by norm_num) hμ))) |
| have hηline : η ≤ d / 8 := min_le_left _ _ |
| have hηneg : η ≤ 3 / 16 := (min_le_right _ _).trans (min_le_left _ _) |
| have hηtail : η ≤ (5 / 16) * ε := |
| (min_le_right _ _).trans ((min_le_right _ _).trans (min_le_left _ _)) |
| have hηmixed : η ≤ 5 / 32 := |
| (min_le_right _ _).trans ((min_le_right _ _).trans |
| ((min_le_right _ _).trans (min_le_left _ _))) |
| have hηcompact : η ≤ μ := |
| (min_le_right _ _).trans ((min_le_right _ _).trans |
| ((min_le_right _ _).trans (min_le_right _ _))) |
| have hordered : ∀ s t : ℝ, d ≤ t - s → η ≤ ‖γ t - γ s‖ := by |
| intro s t hgap |
| have hst : s ≤ t := by linarith |
| by_cases ht1 : t ≤ 1 |
| · rw [hline s t hst ht1] |
| exact hηline.trans (by linarith) |
| have ht1' : 1 < t := lt_of_not_ge ht1 |
| by_cases hsneg : s ≤ -4 |
| · exact hηneg.trans (hnegative s t hsneg ht1') |
| by_cases hs3 : 3 ≤ s |
| · have hsq : ((5 / 16 : ℝ) * ε) ^ 2 ≤ ‖γ t - γ s‖ ^ 2 := by |
| rw [htail s t hs3 hst] |
| have hepssq : ε ^ 2 ≤ 2 * ε := by |
| nlinarith [mul_le_mul_of_nonneg_right hεhalf hε.le] |
| have hgap' := hsinc (t - s) hgap |
| nlinarith |
| have hlow : (5 / 16 : ℝ) * ε ≤ ‖γ t - γ s‖ := |
| (sq_le_sq₀ (mul_nonneg (by norm_num) hε.le) (norm_nonneg _)).mp hsq |
| exact hηtail.trans hlow |
| by_cases ht5 : 5 ≤ t |
| · have hsq : (5 / 32 : ℝ) ^ 2 ≤ ‖γ t - γ s‖ ^ 2 := by |
| have hm := hmixed s t (le_of_not_ge hs3) ht5 |
| nlinarith |
| have hlow : (5 / 32 : ℝ) ≤ ‖γ t - γ s‖ := |
| (sq_le_sq₀ (by norm_num) (norm_nonneg _)).mp hsq |
| exact hηmixed.trans hlow |
| · exact hηcompact.trans (hcompact s ⟨by linarith, by linarith⟩ |
| t ⟨by linarith, by linarith⟩ hgap) |
| refine ⟨η, hηpos, ?_⟩ |
| intro s t hgap |
| by_cases hst : s ≤ t |
| · exact hordered s t (by simpa [abs_of_nonneg (sub_nonneg.mpr hst)] using hgap) |
| · have hts : t ≤ s := (lt_of_not_ge hst).le |
| have hgap' : d ≤ s - t := by |
| simpa [abs_of_nonpos (sub_nonpos.mpr hts), neg_sub] using hgap |
| simpa only [norm_sub_rev] using hordered t s hgap' |
|
|
| end ThompsonNonamenability.Analytic |
|
|
| end |
|
|
| end OAI |
|
|
