| import OAI.Analysis.FiniteFactor.Sampling.Kernel |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory Finset |
| noncomputable section |
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| theorem samplingCharacter_nat (r : ℕ) (t : ℝ) : |
| samplingCharacter (r : ℤ) t = samplingCharacter 1 t ^ r := by |
| induction r with |
| | zero => simp [samplingCharacter_zero] |
| | succ r ih => |
| rw [Nat.cast_add, Nat.cast_one, samplingCharacter_add, ih, pow_succ] |
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| / |
| theorem samplingKernel_geometric_bound (H : ℕ) (t : ℝ) : |
| ‖∑ r ∈ Finset.range H, samplingCharacter (r : ℤ) t‖ * |
| ‖samplingCharacter 1 t - 1‖ ≤ 2 := by |
| simp_rw [samplingCharacter_nat] |
| rw [← norm_mul, geom_sum_mul] |
| calc |
| _ ≤ ‖samplingCharacter 1 t ^ H‖ + ‖(1 : ℂ)‖ := norm_sub_le _ _ |
| _ = _ := by norm_num [norm_pow, samplingCharacter_norm] |
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| / |
| theorem samplingCharacter_chord_lower (δ t : ℝ) (hδ : 0 ≤ δ) |
| (ht : δ ≤ t) (ht' : t ≤ 1-δ) : |
| 4*δ ≤ ‖samplingCharacter 1 t-1‖ := by |
| have he : samplingCharacter 1 t = Complex.exp (Complex.I * ((2*Real.pi*t : ℝ) : ℂ)) := by |
| unfold samplingCharacter |
| congr 1 |
| push_cast |
| ring |
| rw [he, Complex.norm_exp_I_mul_ofReal_sub_one] |
| have harg : 2*Real.pi*t/2 = Real.pi*t := by ring |
| rw [harg, Real.norm_eq_abs, abs_mul, show |(2 : ℝ)| = 2 by norm_num] |
| have hsin : 2*δ ≤ Real.sin (Real.pi*t) := by |
| by_cases hhalf : t ≤ 1/2 |
| · have ht0 : 0 ≤ t := hδ.trans ht |
| have hh := Real.mul_le_sin (x := Real.pi*t) (by positivity) |
| (by nlinarith [Real.pi_pos]) |
| have he : 2/Real.pi*(Real.pi*t) = 2*t := by field_simp |
| rw [he] at hh |
| linarith |
| · have ht0 : 0 ≤ 1-t := by linarith |
| have hh := Real.mul_le_sin (x := Real.pi*(1-t)) (by positivity) |
| (by nlinarith [Real.pi_pos]) |
| have he : 2/Real.pi*(Real.pi*(1-t)) = 2*(1-t) := by field_simp |
| rw [he, show Real.pi*(1-t) = Real.pi-Real.pi*t by ring, Real.sin_pi_sub] at hh |
| linarith |
| have := le_abs_self (Real.sin (Real.pi*t)) |
| linarith |
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|
| theorem samplingKernel_tail_bound (H : ℕ) (hH : 0 < H) (δ t : ℝ) |
| (hδ : 0 < δ) (ht : δ ≤ t) (ht' : t ≤ 1-δ) : |
| samplingKernel H t ≤ 1/(4*(H : ℝ)*δ^2) := by |
| have hc := samplingCharacter_chord_lower δ t hδ.le ht ht' |
| have hg := samplingKernel_geometric_bound H t |
| let A := ‖∑ r ∈ Finset.range H, samplingCharacter (r : ℤ) t‖ |
| have hA : 0 ≤ A := norm_nonneg _ |
| have hbound : A ≤ 1/(2*δ) := by |
| apply (le_div_iff₀ (by positivity : 0 < 2*δ)).mpr |
| have hh := mul_le_mul_of_nonneg_left hc hA |
| dsimp [A] at * |
| nlinarith |
| have hsq := pow_le_pow_left₀ hA hbound 2 |
| have hHR : 0 < (H : ℝ) := by exact_mod_cast hH |
| unfold samplingKernel |
| calc |
| _ ≤ (1/(2*δ))^2/(H : ℝ) := div_le_div_of_nonneg_right hsq hHR.le |
| _ = _ := by ring |
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| / |
| of the integer cut. -/ |
| theorem samplingKernel_tail_integral (H : ℕ) (hH : 0 < H) (δ : ℝ) (hδ : 0 < δ) : |
| (∫ t in Set.Icc δ (1-δ), samplingKernel H t ∂unitIntervalMeasure) ≤ |
| 1/(4*(H : ℝ)*δ^2) := by |
| have hki : Integrable (samplingKernel H) unitIntervalMeasure := |
| (samplingKernel_continuous H).integrableOn_Icc.mono_set Set.Ico_subset_Icc_self |
| rw [← integral_indicator measurableSet_Icc] |
| have hle : ∀ t, (Set.Icc δ (1-δ)).indicator (samplingKernel H) t ≤ 1/(4*(H : ℝ)*δ^2) := by |
| intro t |
| by_cases ht : t ∈ Set.Icc δ (1-δ) |
| · rw [Set.indicator_of_mem ht] |
| exact samplingKernel_tail_bound H hH δ t hδ ht.1 ht.2 |
| · rw [Set.indicator_of_notMem ht] |
| positivity |
| have hh := integral_mono_ae (hki.indicator measurableSet_Icc) (integrable_const (1/(4*(H : ℝ)*δ^2))) |
| (Filter.Eventually.of_forall hle) |
| simpa using hh |
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| end |
| end FiniteFactor |
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| end OAI |
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