| 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 | |