| import Mathlib.Algebra.Order.BigOperators.Group.Finset |
| import Mathlib.Data.Finset.Interval |
| import Mathlib.Tactic.FieldSimp |
| import OAI.NumberTheory.PiExponent.Analysis.DyadicPowers |
| import OAI.NumberTheory.PiExponent.Analysis.DyadicSummation |
| import OAI.NumberTheory.PiExponent.Analysis.FiniteSpacing |
| import OAI.NumberTheory.PiExponent.Analysis.FlintHills |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| open scoped BigOperators |
|
|
| namespace PiExponent |
|
|
| def centeredResidue (alpha : ℝ) (q : ℕ) : ℝ := |
| (q : ℝ) * alpha - (round ((q : ℝ) * alpha) : ℝ) |
|
|
| theorem abs_centeredResidue (alpha : ℝ) (q : ℕ) : |
| |centeredResidue alpha q| = integerDistance ((q : ℝ) * alpha) := rfl |
|
|
| theorem integerDistance_sub_le (x y : ℝ) : |
| integerDistance (x - y) ≤ |(x - (round x : ℝ)) - (y - (round y : ℝ))| := by |
| have h := integerDistance_le (x - y) (round x - round y) |
| push_cast at h |
| convert h using 1 |
| congr 1 |
| ring |
|
|
| theorem centeredResidue_separation {alpha c nu : ℝ} (hc : 0 < c) (hnu : 1 < nu) |
| (hlower : ∀ q : ℕ, 0 < q → |
| c * (q : ℝ) ^ (1 - nu) ≤ integerDistance ((q : ℝ) * alpha)) |
| {K q r : ℕ} (hq : q ∈ Finset.Ico K (2 * K)) |
| (hr : r ∈ Finset.Ico K (2 * K)) (hqr : q < r) : |
| c * (2 * (K : ℝ)) ^ (1 - nu) ≤ |
| |centeredResidue alpha r - centeredResidue alpha q| := by |
| have hdiff : 0 < r - q := Nat.sub_pos_of_lt hqr |
| have hdiffR : (0 : ℝ) < (r - q : ℕ) := by exact_mod_cast hdiff |
| have hdiffle : ((r - q : ℕ) : ℝ) ≤ 2 * (K : ℝ) := by |
| have hqK := (Finset.mem_Ico.mp hq).1 |
| have hrK := (Finset.mem_Ico.mp hr).2 |
| have hlt : r - q < K := by omega |
| exact_mod_cast hlt.le.trans (by omega : K ≤ 2 * K) |
| have hpower := Real.rpow_le_rpow_of_nonpos hdiffR hdiffle (by linarith : 1 - nu ≤ 0) |
| have hsep := integerDistance_sub_le ((r : ℝ) * alpha) ((q : ℝ) * alpha) |
| have hcast : ((r - q : ℕ) : ℝ) * alpha = (r : ℝ) * alpha - (q : ℝ) * alpha := by |
| rw [Nat.cast_sub hqr.le] |
| ring |
| calc |
| c * (2 * (K : ℝ)) ^ (1 - nu) ≤ c * ((r - q : ℕ) : ℝ) ^ (1 - nu) := |
| mul_le_mul_of_nonneg_left hpower hc.le |
| _ ≤ integerDistance (((r - q : ℕ) : ℝ) * alpha) := hlower (r - q) hdiff |
| _ ≤ |centeredResidue alpha r - centeredResidue alpha q| := by |
| rw [hcast] |
| exact hsep |
|
|
| theorem integerDistance_block_lower {alpha c nu : ℝ} (hc : 0 < c) (hnu : 1 < nu) |
| (hlower : ∀ q : ℕ, 0 < q → |
| c * (q : ℝ) ^ (1 - nu) ≤ integerDistance ((q : ℝ) * alpha)) |
| {K q : ℕ} (hK : 0 < K) (hq : q ∈ Finset.Ico K (2 * K)) : |
| c * (2 * (K : ℝ)) ^ (1 - nu) ≤ integerDistance ((q : ℝ) * alpha) := by |
| have hqpos : 0 < q := hK.trans_le (Finset.mem_Ico.mp hq).1 |
| have hqR : (0 : ℝ) < q := by exact_mod_cast hqpos |
| have hqle : (q : ℝ) ≤ 2 * (K : ℝ) := by exact_mod_cast (Finset.mem_Ico.mp hq).2.le |
| exact (mul_le_mul_of_nonneg_left |
| (Real.rpow_le_rpow_of_nonpos hqR hqle (by linarith : 1 - nu ≤ 0)) hc.le).trans |
| (hlower q hqpos) |
|
|
| theorem integerDistance_block_sum {alpha c nu : ℝ} (hc : 0 < c) (hnu : 1 < nu) |
| (hlower : ∀ q : ℕ, 0 < q → |
| c * (q : ℝ) ^ (1 - nu) ≤ integerDistance ((q : ℝ) * alpha)) |
| {K : ℕ} (hK : 0 < K) : |
| (∑ q ∈ Finset.Ico K (2 * K), 1 / (integerDistance ((q : ℝ) * alpha)) ^ 2) ≤ |
| 4 / (c * (2 * (K : ℝ)) ^ (1 - nu)) ^ 2 := by |
| classical |
| let rows := Finset.Ico K (2 * K) |
| let d : ℝ := c * (2 * (K : ℝ)) ^ (1 - nu) |
| have hd : 0 < d := by dsimp [d]; positivity |
| have hmin : ∀ q ∈ rows, d ≤ integerDistance ((q : ℝ) * alpha) := |
| fun q hq => integerDistance_block_lower hc hnu hlower hK hq |
| have hsep : ∀ q ∈ rows, ∀ r ∈ rows, q ≠ r → |
| d ≤ |centeredResidue alpha q - centeredResidue alpha r| := by |
| intro q hq r hr hqr |
| rcases lt_or_gt_of_ne hqr with hqr | hrq |
| · simpa only [abs_sub_comm] using centeredResidue_separation hc hnu hlower hq hr hqr |
| · exact centeredResidue_separation hc hnu hlower hr hq hrq |
| have hpos := FiniteSpacing.sum_inv_sq_le_two_div_sq_indexed |
| (rows.filter fun q => 0 ≤ centeredResidue alpha q) |
| (fun q => integerDistance ((q : ℝ) * alpha)) hd |
| (fun q hq => hmin q (Finset.mem_filter.mp hq).1) |
| (by |
| intro q hq r hr hqr |
| change d ≤ abs (abs (centeredResidue alpha q) - abs (centeredResidue alpha r)) |
| rw [abs_of_nonneg (Finset.mem_filter.mp hq).2, |
| abs_of_nonneg (Finset.mem_filter.mp hr).2] |
| exact hsep q (Finset.mem_filter.mp hq).1 r (Finset.mem_filter.mp hr).1 hqr) |
| have hneg := FiniteSpacing.sum_inv_sq_le_two_div_sq_indexed |
| (rows.filter fun q => ¬ 0 ≤ centeredResidue alpha q) |
| (fun q => integerDistance ((q : ℝ) * alpha)) hd |
| (fun q hq => hmin q (Finset.mem_filter.mp hq).1) |
| (by |
| intro q hq r hr hqr |
| change d ≤ abs (abs (centeredResidue alpha q) - abs (centeredResidue alpha r)) |
| rw [abs_of_neg (lt_of_not_ge (Finset.mem_filter.mp hq).2), |
| abs_of_neg (lt_of_not_ge (Finset.mem_filter.mp hr).2), neg_sub_neg, abs_sub_comm] |
| exact hsep q (Finset.mem_filter.mp hq).1 r (Finset.mem_filter.mp hr).1 hqr) |
| change (∑ q ∈ rows, 1 / (integerDistance ((q : ℝ) * alpha)) ^ 2) ≤ 4 / d ^ 2 |
| rw [← Finset.sum_filter_add_sum_filter_not rows (fun q => 0 ≤ centeredResidue alpha q)] |
| calc |
| _ ≤ 2 / d ^ 2 + 2 / d ^ 2 := add_le_add hpos hneg |
| _ = _ := by ring |
|
|
| theorem inverse_cube_integerDistance_block_sum {alpha c nu : ℝ} |
| (hc : 0 < c) (hnu : 1 < nu) |
| (hlower : ∀ q : ℕ, 0 < q → |
| c * (q : ℝ) ^ (1 - nu) ≤ integerDistance ((q : ℝ) * alpha)) |
| {K : ℕ} (hK : 0 < K) : |
| (∑ q ∈ Finset.Ico K (2 * K), |
| 1 / ((q : ℝ) ^ 3 * (integerDistance ((q : ℝ) * alpha)) ^ 2)) ≤ |
| 4 / ((K : ℝ) ^ 3 * (c * (2 * (K : ℝ)) ^ (1 - nu)) ^ 2) := by |
| have hKR : (0 : ℝ) < K := by exact_mod_cast hK |
| calc |
| _ ≤ ∑ q ∈ Finset.Ico K (2 * K), |
| (1 / (K : ℝ) ^ 3) * (1 / (integerDistance ((q : ℝ) * alpha)) ^ 2) := by |
| apply Finset.sum_le_sum |
| intro q hq |
| have hKq : (K : ℝ) ≤ q := by exact_mod_cast (Finset.mem_Ico.mp hq).1 |
| have hdist : 0 < integerDistance ((q : ℝ) * alpha) := |
| (mul_pos hc (Real.rpow_pos_of_pos (by positivity) _)).trans_le |
| (integerDistance_block_lower hc hnu hlower hK hq) |
| rw [one_div_mul_one_div] |
| exact div_le_div_of_nonneg_left (by norm_num) (by positivity) |
| (mul_le_mul_of_nonneg_right (pow_le_pow_left₀ hKR.le hKq 3) (sq_nonneg _)) |
| _ = (1 / (K : ℝ) ^ 3) * |
| ∑ q ∈ Finset.Ico K (2 * K), 1 / (integerDistance ((q : ℝ) * alpha)) ^ 2 := by |
| rw [Finset.mul_sum] |
| _ ≤ (1 / (K : ℝ) ^ 3) * (4 / (c * (2 * (K : ℝ)) ^ (1 - nu)) ^ 2) := |
| mul_le_mul_of_nonneg_left (integerDistance_block_sum hc hnu hlower hK) (by positivity) |
| _ = _ := by ring |
|
|
| theorem summable_inverse_cube_integerDistance {alpha c nu : ℝ} |
| (hc : 0 < c) (hnu1 : 1 < nu) (hnu2 : nu < 5 / 2) |
| (hlower : ∀ q : ℕ, 0 < q → |
| c * (q : ℝ) ^ (1 - nu) ≤ integerDistance ((q : ℝ) * alpha)) : |
| Summable (fun q : ℕ => |
| 1 / ((q : ℝ) ^ 3 * (integerDistance ((q : ℝ) * alpha)) ^ 2)) := by |
| apply summable_of_dyadic_blocks (fun q hq => by positivity) |
| (summable_dyadic_block_bound hc hnu2) |
| intro k |
| have hblock := inverse_cube_integerDistance_block_sum hc hnu1 hlower |
| (pow_pos (by norm_num : (0 : ℕ) < 2) k) |
| have hend : 2 * 2 ^ k = 2 ^ (k + 1) := by rw [pow_succ, Nat.mul_comm] |
| rw [hend] at hblock |
| simpa only [Nat.cast_pow, Nat.cast_ofNat] using hblock |
|
|
| end PiExponent |
|
|
| end |
|
|
| end OAI |
|
|
