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180bc03 | 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 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 | import OAI.NumberTheory.TwoPoint.ShortIntervals.MRTWindowBounds
namespace OAI
/-! Bounded measurable finite interval sums and the squared two-window
inequality. These statements allow arbitrary complex coefficients. -/
namespace TwoPointCorrelations
open MeasureTheory Finset Set
open scoped Classical
lemma mrt_interval_sum_measurable {α : Type*} [MeasurableSpace α]
(S : Finset ℕ) (a : ℕ → ℂ) {x h : α → ℝ}
(hx : Measurable x) (hh : Measurable h) :
Measurable (fun t => mrtIntervalSum S a (x t) (h t)) := by
apply Finset.measurable_sum
intro n _
exact Measurable.ite
((measurableSet_lt hx measurable_const).inter
(measurableSet_le measurable_const (hx.add hh)))
measurable_const measurable_const
lemma mrt_interval_sum_norm (S : Finset ℕ) (a : ℕ → ℂ) (x h : ℝ) :
‖mrtIntervalSum S a x h‖ ≤ ∑ n ∈ S, ‖a n‖ := by
unfold mrtIntervalSum
apply (norm_sum_le _ _).trans
apply sum_le_sum
intro n _
split_ifs
· exact le_rfl
· simpa only [norm_zero] using (norm_nonneg (a n))
lemma mrt_interval_sum_sq_integrableOn {α : Type*} [MeasurableSpace α]
(S : Finset ℕ) (a : ℕ → ℂ) {μ : Measure α} {s : Set α}
(hs : μ s ≠ ⊤) {x h : α → ℝ} (hx : Measurable x) (hh : Measurable h) :
IntegrableOn (fun t => ‖mrtIntervalSum S a (x t) (h t)‖ ^ 2) s μ := by
apply Measure.integrableOn_of_bounded hs
(((mrt_interval_sum_measurable S a hx hh).norm.pow_const 2).aestronglyMeasurable)
apply Filter.Eventually.of_forall
intro t
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
exact pow_le_pow_left₀ (norm_nonneg _) (mrt_interval_sum_norm S a (x t) (h t)) 2
lemma mrt_interval_sum_sq_intervalIntegrable (S : Finset ℕ) (a : ℕ → ℂ)
{x h : ℝ → ℝ} (hx : Measurable x) (hh : Measurable h) (b c : ℝ) :
IntervalIntegrable (fun t => ‖mrtIntervalSum S a (x t) (h t)‖ ^ 2)
volume b c := by
rw [intervalIntegrable_iff]
apply mrt_interval_sum_sq_integrableOn S a _ hx hh
exact measure_Ioc_lt_top.ne
lemma mrt_interval_sum_sq_split (S : Finset ℕ) (a : ℕ → ℂ) {x h w : ℝ}
(hh : 0 ≤ h) (hw : h ≤ w) :
‖mrtIntervalSum S a x h‖ ^ 2 ≤
2 * ‖mrtIntervalSum S a x w‖ ^ 2 +
2 * ‖mrtIntervalSum S a (x + h) (w - h)‖ ^ 2 := by
rw [← mrt_interval_sum_difference S a hh hw]
have hn := norm_sub_le (mrtIntervalSum S a x w)
(mrtIntervalSum S a (x + h) (w - h))
have hs := pow_le_pow_left₀ (norm_nonneg _) hn 2
nlinarith [sq_nonneg (‖mrtIntervalSum S a x w‖ -
‖mrtIntervalSum S a (x + h) (w - h)‖)]
theorem mrt_interval_sum_averaged_sq (S : Finset ℕ) (a : ℕ → ℂ)
(x : ℝ) {h : ℝ} (hh : 0 < h) :
‖mrtIntervalSum S a x h‖ ^ 2 ≤ h⁻¹ *
((∫ w in h..3 * h, ‖mrtIntervalSum S a x w‖ ^ 2) +
∫ w in h..3 * h, ‖mrtIntervalSum S a (x + h) (w - h)‖ ^ 2) := by
have hi₁ := mrt_interval_sum_sq_intervalIntegrable S a (x := fun _ => x) (h := id)
measurable_const measurable_id h (3 * h)
have hi₂ := mrt_interval_sum_sq_intervalIntegrable S a
(x := fun _ => x+h) (h := fun w => w-h)
measurable_const (measurable_id.sub measurable_const) h (3 * h)
have hm := intervalIntegral.integral_mono_on (show h ≤ 3 * h by linarith)
(intervalIntegrable_const : IntervalIntegrable
(fun _ : ℝ => ‖mrtIntervalSum S a x h‖ ^ 2) volume h (3 * h))
((hi₁.const_mul 2).add (hi₂.const_mul 2))
(fun w hw => mrt_interval_sum_sq_split S a hh.le hw.1)
rw [intervalIntegral.integral_const, intervalIntegral.integral_add
(hi₁.const_mul 2) (hi₂.const_mul 2),
intervalIntegral.integral_const_mul, intervalIntegral.integral_const_mul] at hm
simp only [smul_eq_mul, id_eq] at hm
apply (le_inv_mul_iff₀ hh).mpr
nlinarith
end TwoPointCorrelations
end OAI
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