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8769036 | 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 | import OAI.NumberTheory.TwoPoint.Fourier.MinorArcPacking
import Mathlib.NumberTheory.Harmonic.Bounds
namespace OAI
/-! Summing a truncated reciprocal over separated real representatives. -/
namespace TwoPointCorrelations
open Finset
open scoped Classical
theorem minor_arc_packing_sum {ι : Type*} (S : Finset ι) (x w : ι → ℝ)
(δ V R : ℝ) (hδ : 0 < δ) (hV : 0 ≤ V)
(hsep : ∀ i ∈ S, ∀ j ∈ S, i ≠ j → δ ≤ |x i - x j|)
(hx : ∀ i ∈ S, |x i| ≤ R) (hw : ∀ i ∈ S, w i ≤ V)
(hwinv : ∀ i ∈ S, 0 < |x i| → w i ≤ 1 / |x i|) :
(∑ i ∈ S, w i) ≤ 2 * V + (2 / δ) * (1 + Real.log (⌈R / δ⌉₊ : ℝ)) := by
let L := ⌈R / δ⌉₊
let b := fun i => ⌊|x i| / δ⌋₊
let F := fun j : ℕ => S.filter (fun i => b i = j)
have hindex (i : ι) (hi : i ∈ S) : b i ≤ L := by
apply Nat.floor_le_of_le
exact (div_le_div_of_nonneg_right (hx i hi) hδ.le).trans (Nat.le_ceil _)
have hband (j : ℕ) (i : ι) (hi : i ∈ F j) :
(j : ℝ) * δ ≤ |x i| ∧ |x i| < (j : ℝ) * δ + δ := by
have he : b i = j := (mem_filter.mp hi).2
have hl := Nat.floor_le (div_nonneg (abs_nonneg (x i)) hδ.le)
have hu := Nat.lt_floor_add_one (|x i| / δ)
change ((b i : ℕ) : ℝ) ≤ |x i| / δ at hl
change |x i| / δ < (b i : ℝ) + 1 at hu
rw [he] at hl hu
constructor
· exact (le_div_iff₀ hδ).mp hl
· have hh := (div_lt_iff₀ hδ).mp hu
nlinarith
have hcard (j : ℕ) : (F j).card ≤ 2 := by
refine (card_le_card (s := F j) (t := S.filter (fun i =>
(j : ℝ) * δ ≤ |x i| ∧ |x i| < (j : ℝ) * δ + δ)) ?_).trans
(minor_arc_absolute_band_packing S x δ ((j : ℝ) * δ) hsep)
intro i hi
exact mem_filter.mpr ⟨(mem_filter.mp hi).1, hband j i hi⟩
have hzero : (∑ i ∈ F 0, w i) ≤ 2 * V := by
calc
_ ≤ ∑ _i ∈ F 0, V := sum_le_sum (fun i hi => hw i (mem_filter.mp hi).1)
_ = ((F 0).card : ℝ) * V := by simp
_ ≤ 2 * V := mul_le_mul_of_nonneg_right (by exact_mod_cast hcard 0) hV
have hpiece (j : ℕ) (hj : j ∈ Icc 1 L) : (∑ i ∈ F j, w i) ≤ 2 / (δ * j) := by
have hj0 : (0 : ℝ) < j := by exact_mod_cast (mem_Icc.mp hj).1
have hbound (i : ι) (hi : i ∈ F j) : w i ≤ 1 / (δ * j) := by
have hlo := (hband j i hi).1
have hpos : 0 < |x i| := (mul_pos hj0 hδ).trans_le hlo
apply (hwinv i (mem_filter.mp hi).1 hpos).trans
exact one_div_le_one_div_of_le (mul_pos hδ hj0) (by nlinarith)
calc
_ ≤ ∑ _i ∈ F j, 1 / (δ * j) := sum_le_sum hbound
_ = ((F j).card : ℝ) * (1 / (δ * j)) := by simp
_ ≤ 2 * (1 / (δ * j)) :=
mul_le_mul_of_nonneg_right (by exact_mod_cast hcard j) (by positivity)
_ = _ := by ring
have hpartition : (∑ i ∈ S, w i) = ∑ j ∈ Icc 0 L, ∑ i ∈ F j, w i := by
symm
exact sum_fiberwise_of_maps_to (fun i hi => mem_Icc.mpr ⟨Nat.zero_le _, hindex i hi⟩) _
have hsplit : (∑ j ∈ Icc 0 L, ∑ i ∈ F j, w i) =
(∑ i ∈ F 0, w i) + ∑ j ∈ Icc 1 L, ∑ i ∈ F j, w i := by
have he : Icc 0 L = insert 0 (Icc 1 L) := by
ext j
simp only [mem_Icc, mem_insert]
omega
rw [he, sum_insert (by simp)]
have hharm : (∑ j ∈ Icc 1 L, 2 / (δ * j)) ≤
(2 / δ) * (1 + Real.log (L : ℝ)) := by
calc
_ = (2 / δ) * (harmonic L : ℝ) := by
rw [harmonic_eq_sum_Icc]
push_cast
rw [mul_sum]
apply sum_congr rfl
intro j _
ring
_ ≤ _ := mul_le_mul_of_nonneg_left (harmonic_le_one_add_log L) (by positivity)
rw [hpartition, hsplit]
exact add_le_add hzero ((sum_le_sum hpiece).trans hharm)
end TwoPointCorrelations
end OAI
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