| import OAI.Analysis.IrrationalRotation.Sampling.CirclePhaseSummability |
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| namespace OAI |
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| noncomputable section |
| open Filter |
| open scoped BigOperators Topology |
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| namespace Invariant233.CircleParameterLaws |
| variable {θ : ℝ} {p : CircleParameterData} (hp : CircleParameterLaws θ p) |
| include hp |
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|
| theorem block_phase_tail_bound (j : ℕ) (hj : 1 ≤ j) : |
| (∑' n : ℕ, ∑ i ∈ p.block (j + 1 + n), p.alpha θ i) ≤ |
| 4 / (p.K (j + 1) : ℝ) := by |
| have hs : Summable (fun n : ℕ => |
| ∑ i ∈ p.block (j + 1 + n), p.alpha θ i) := by |
| have h := (summable_nat_add_iff j).mpr hp.summable_block_phases |
| simpa only [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using h |
| have hK : Summable (fun n : ℕ => 1 / (p.K (j + 1 + n) : ℝ)) := by |
| have h := (summable_nat_add_iff j).mpr hp.summable_reciprocal |
| simpa only [Nat.add_assoc, Nat.add_comm, Nat.add_left_comm] using h |
| calc |
| _ ≤ ∑' n : ℕ, 2 * (1 / (p.K (j + 1 + n) : ℝ)) := by |
| apply Summable.tsum_le_tsum _ hs (hK.mul_left 2) |
| intro n |
| simpa only [mul_one_div] using hp.block_phase_sum_le (j + 1 + n) (by omega) |
| _ = 2 * ∑' n : ℕ, 1 / (p.K (j + 1 + n) : ℝ) := tsum_mul_left |
| _ ≤ 2 * (2 / (p.K (j + 1) : ℝ)) := |
| mul_le_mul_of_nonneg_left (hp.reciprocal_tail_bound j hj) (by norm_num) |
| _ = _ := by ring |
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| theorem sampled_phase_tail_bound (j r n : ℕ) (hj : 1 ≤ j) |
| (hr : r < p.q (p.blockEnd (j - 1))) (hn : n ∈ p.samples j r) : |
| (n : ℝ) * (∑' s : ℕ, ∑ i ∈ p.block (j + 1 + s), p.alpha θ i) ≤ |
| 4 * p.Q j / (p.K (j + 1) : ℝ) := by |
| have hK : 0 < (p.K (j + 1) : ℝ) := by |
| exact_mod_cast hp.K_pos (j + 1) (by omega) |
| calc |
| _ ≤ (n : ℝ) * (4 / (p.K (j + 1) : ℝ)) := |
| mul_le_mul_of_nonneg_left (hp.block_phase_tail_bound j hj) (Nat.cast_nonneg n) |
| _ ≤ p.Q j * (4 / (p.K (j + 1) : ℝ)) := |
| mul_le_mul_of_nonneg_right (hp.samples_range j hj r hr n hn).2 (by positivity) |
| _ = _ := by ring |
|
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| theorem early_trimming_bound (j : ℕ) : |
| (p.blockEnd (j - 1) : ℝ) * (p.q (p.blockEnd (j - 1)) : ℝ) * p.ell j ≤ |
| 1 / ((j + 1 : ℕ) : ℝ)^2 := by |
| have hq : 0 < (p.q (p.blockEnd (j - 1)) : ℝ) := by |
| exact_mod_cast hp.q_pos (p.blockEnd (j - 1)) |
| have ha : 0 ≤ (p.blockEnd (j - 1) : ℝ) := Nat.cast_nonneg _ |
| have hr : (p.blockEnd (j - 1) : ℝ) / (1 + (p.blockEnd (j - 1) : ℝ)) ≤ 1 := by |
| apply (div_le_one (by positivity)).mpr |
| linarith |
| calc |
| _ = ((p.blockEnd (j - 1) : ℝ) / (1 + (p.blockEnd (j - 1) : ℝ))) * |
| (1 / ((j + 1 : ℕ) : ℝ)^2) := by |
| unfold CircleParameterData.ell |
| field_simp |
| _ ≤ 1 * (1 / ((j + 1 : ℕ) : ℝ)^2) := |
| mul_le_mul_of_nonneg_right hr (by positivity) |
| _ = _ := one_mul _ |
|
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| theorem discarded_mass_bound (j : ℕ) (hj : 1 ≤ j) : |
| (p.blockEnd (j - 1) : ℝ) * (p.q (p.blockEnd (j - 1)) : ℝ) * p.ell j + |
| 2 * p.Q j * (∑' n : ℕ, 1 / (p.K (j + 1 + n) : ℝ)) ≤ |
| 1 / ((j + 1 : ℕ) : ℝ)^2 + 4 * p.Q j / (p.K (j + 1) : ℝ) := by |
| apply add_le_add (hp.early_trimming_bound j) |
| have hQ : 0 ≤ p.Q j := Real.rpow_nonneg (Nat.cast_nonneg _) _ |
| calc |
| _ ≤ 2 * p.Q j * (2 / (p.K (j + 1) : ℝ)) := |
| mul_le_mul_of_nonneg_left (hp.reciprocal_tail_bound j hj) (by positivity) |
| _ = _ := by ring |
|
|
| theorem tendsto_discarded_mass_bound : |
| Tendsto (fun n : ℕ => 1 / ((n + 2 : ℕ) : ℝ)^2 + |
| 4 * p.Q (n + 1) / (p.K (n + 2) : ℝ)) atTop (𝓝 0) := by |
| have h : Tendsto (fun n : ℕ => 1 / ((n + 2 : ℕ) : ℝ)) atTop (𝓝 (0 : ℝ)) := by |
| exact (tendsto_add_atTop_iff_nat 2).mpr tendsto_one_div_atTop_nhds_zero_nat |
| have hsq : Tendsto (fun n : ℕ => 1 / ((n + 2 : ℕ) : ℝ)^2) |
| atTop (𝓝 (0 : ℝ)) := by |
| simpa only [one_div_pow, zero_pow (by decide : 2 ≠ 0)] using h.pow 2 |
| have hg := hsq.add (hp.tendsto_sampling_horizon.const_mul 4) |
| simpa only [mul_div_assoc, zero_add, mul_zero] using hg |
|
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| end Invariant233.CircleParameterLaws |
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| end |
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| end OAI |
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