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b17ead8 | 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 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 | import OAI.MeasureTheory.Falconer.Angular.AngularGainFinite
import Mathlib.Data.Int.Interval
namespace OAI
/-!
# A common line grid for the planar incidence obstruction
Each original strip gives a bounded list of common-grid line parameters.
The list is attached to its original direction label. Rounding therefore
turns the manuscript's covering-number upper bound into the cardinality
upper bound needed by `published_furstenberg_labelled_obstruction`.
-/
namespace Falconer
open Set
noncomputable def gridLine (δ : ℝ) (p : ℤ × ℤ) : Space 2 :=
WithLp.toLp 2 ![δ * (p.1 : ℝ), δ * (p.2 : ℝ)]
@[simp] theorem gridLine_fst (δ : ℝ) (p : ℤ × ℤ) :
gridLine δ p 0 = δ * (p.1 : ℝ) := rfl
@[simp] theorem gridLine_snd (δ : ℝ) (p : ℤ × ℤ) :
gridLine δ p 1 = δ * (p.2 : ℝ) := rfl
private theorem integer_grid_separation {δ : ℝ} (hδ : 0 < δ) {i j : ℤ}
(hij : i ≠ j) : δ ≤ |δ * (i : ℝ) - δ * (j : ℝ)| := by
have hzpos : (0 : ℤ) < |i - j| := abs_pos.mpr (sub_ne_zero.mpr hij)
have hz : (1 : ℤ) ≤ |i - j| := by omega
have hr : (1 : ℝ) ≤ |(i : ℝ) - (j : ℝ)| := by exact_mod_cast hz
calc
δ = δ * 1 := (mul_one _).symm
_ ≤ δ * |(i : ℝ) - (j : ℝ)| := mul_le_mul_of_nonneg_left hr hδ.le
_ = |δ * (i : ℝ) - δ * (j : ℝ)| := by
rw [← mul_sub, abs_mul, abs_of_pos hδ]
theorem gridLine_dist_ge {δ : ℝ} (hδ : 0 < δ) {p q : ℤ × ℤ} (hpq : p ≠ q) :
δ ≤ dist (gridLine δ p) (gridLine δ q) := by
by_cases hfst : p.1 = q.1
· have hsnd : p.2 ≠ q.2 := by
intro h
exact hpq (Prod.ext hfst h)
calc
δ ≤ |δ * (p.2 : ℝ) - δ * (q.2 : ℝ)| := integer_grid_separation hδ hsnd
_ = dist (gridLine δ p 1) (gridLine δ q 1) := by
rw [gridLine_snd, gridLine_snd, Real.dist_eq]
_ ≤ dist (gridLine δ p) (gridLine δ q) := PiLp.dist_apply_le _ _ 1
· calc
δ ≤ |δ * (p.1 : ℝ) - δ * (q.1 : ℝ)| := integer_grid_separation hδ hfst
_ = dist (gridLine δ p 0) (gridLine δ q 0) := by
rw [gridLine_fst, gridLine_fst, Real.dist_eq]
_ ≤ dist (gridLine δ p) (gridLine δ q) := PiLp.dist_apply_le _ _ 0
theorem gridLine_injective {δ : ℝ} (hδ : 0 < δ) : Function.Injective (gridLine δ) := by
intro p q h
by_contra hpq
have hsep := gridLine_dist_ge hδ hpq
rw [h, dist_self] at hsep
exact hδ.not_ge hsep
/-- Every finite subfamily of the common line grid is δ-separated. -/
theorem gridLine_image_separated {δ : ℝ} (hδ : 0 < δ) (S : Finset (ℤ × ℤ)) :
FiniteSeparatedAt δ (S.image (gridLine δ)) := by
classical
intro x hx y hy hxy
obtain ⟨p, _, rfl⟩ := Finset.mem_image.mp hx
obtain ⟨q, _, rfl⟩ := Finset.mem_image.mp hy
apply gridLine_dist_ge hδ
intro hpq
exact hxy (congrArg (gridLine δ) hpq)
theorem roundedIncidentLine_family_separated {ι : Type*} [DecidableEq ι]
{δ : ℝ} (hδ : 0 < δ) (x : Space 2) (m : ι → ℝ) (D : Finset ι) :
FiniteSeparatedAt δ (D.image (fun i => roundedIncidentLine δ x (m i))) := by
intro l hl l' hl' hll'
obtain ⟨i, _, rfl⟩ := Finset.mem_image.mp hl
obtain ⟨j, _, rfl⟩ := Finset.mem_image.mp hl'
apply gridLine_dist_ge hδ
(p := (round (m i / δ), round ((x 1 - lineGridRound δ (m i) * x 0) / δ)))
(q := (round (m j / δ), round ((x 1 - lineGridRound δ (m j) * x 0) / δ)))
intro heq
exact hll' (congrArg (gridLine δ) heq)
/-- A fixed smaller chart leaves room for rounding while staying inside
the unit chart used by the published finite theorem. -/
theorem roundedIncidentLine_in_chart {δ : ℝ} (hδ : 0 < δ) (hδ1 : δ ≤ 1 / 2)
(x : Space 2) (hx0 : |x 0| ≤ 1 / 4) (hx1 : |x 1| ≤ 1 / 4)
(m : ℝ) (hm : |m| ≤ 1 / 2) :
|roundedIncidentLine δ x m 0| ≤ 1 ∧ |roundedIncidentLine δ x m 1| ≤ 1 := by
have hM : |lineGridRound δ m| ≤ 1 := by
have hsum := abs_sub_le (lineGridRound δ m) m 0
have herr := lineGridRound_error hδ m
rw [sub_zero, sub_zero, abs_sub_comm (lineGridRound δ m)] at hsum
linarith
constructor
· exact hM
· have ht : |x 1 - lineGridRound δ m * x 0| ≤ 1 / 2 := by
have hprod : |lineGridRound δ m * x 0| ≤ 1 / 4 := by
rw [abs_mul]
exact (mul_le_mul hM hx0 (abs_nonneg _) (by norm_num)).trans_eq (one_mul _)
have htri : |x 1 - lineGridRound δ m * x 0| ≤
|x 1| + |lineGridRound δ m * x 0| := by
simpa using abs_sub_le (x 1) 0 (lineGridRound δ m * x 0)
exact htri.trans (by linarith)
have hsum := abs_sub_le
(lineGridRound δ (x 1 - lineGridRound δ m * x 0))
(x 1 - lineGridRound δ m * x 0) 0
have herr := lineGridRound_error hδ (x 1 - lineGridRound δ m * x 0)
rw [sub_zero, sub_zero, abs_sub_comm (lineGridRound δ (x 1 - lineGridRound δ m * x 0))]
at hsum
change |lineGridRound δ (x 1 - lineGridRound δ m * x 0)| ≤ 1
linarith
/-- The bounded candidate intercept list for one strip, retaining its
original slope label even if other labels have the same rounded slope. -/
noncomputable def roundedStripCandidates (δ m a : ℝ) (L : ℕ) : Finset (Space 2) :=
(Finset.Icc (round (a / δ) - (L : ℤ)) (round (a / δ) + (L : ℤ))).image
(fun j => gridLine δ (round (m / δ), j))
theorem roundedStripCandidates_card_le (δ m a : ℝ) (L : ℕ) :
(roundedStripCandidates δ m a L).card ≤ 2 * L + 1 := by
classical
calc
(roundedStripCandidates δ m a L).card ≤
(Finset.Icc (round (a / δ) - (L : ℤ)) (round (a / δ) + (L : ℤ))).card :=
Finset.card_image_le
_ = 2 * L + 1 := by
rw [Int.card_Icc]
omega
/-- A rounded line through any point of a strip belongs to that strip's
bounded candidate list. This is the fixed-width loss in `ang:planar-measures`. -/
theorem roundedIncidentLine_mem_candidates {δ C a m : ℝ} (hδ : 0 < δ)
(x : Space 2) (hx : |x 0| ≤ 1)
(hstrip : |x 1 - m * x 0 - a| ≤ C * δ)
(L : ℕ) (hL : C + 2 ≤ (L : ℝ)) :
roundedIncidentLine δ x m ∈ roundedStripCandidates δ m a L := by
classical
let j : ℤ := round ((x 1 - lineGridRound δ m * x 0) / δ)
let j₀ : ℤ := round (a / δ)
have hnear : |δ * (j : ℝ) - a| ≤ (C + 1) * δ :=
roundedIncidentLine_near_strip hδ x hx hstrip
have hcenter : |a - δ * (j₀ : ℝ)| ≤ δ / 2 := lineGridRound_error hδ a
have hsum : |δ * (j : ℝ) - δ * (j₀ : ℝ)| ≤ δ * (L : ℝ) := by
have htri := (abs_sub_le (δ * (j : ℝ)) a (δ * (j₀ : ℝ))).trans
(add_le_add hnear hcenter)
exact htri.trans (by nlinarith)
have habs : |(j : ℝ) - (j₀ : ℝ)| ≤ (L : ℝ) := by
apply (mul_le_mul_iff_right₀ hδ).mp
calc
δ * |(j : ℝ) - (j₀ : ℝ)| = |δ * (j : ℝ) - δ * (j₀ : ℝ)| := by
rw [← mul_sub, abs_mul, abs_of_pos hδ]
_ ≤ δ * (L : ℝ) := hsum
have hint : |j - j₀| ≤ (L : ℤ) := by exact_mod_cast habs
have hbounds := abs_le.mp hint
have hmem : round (a / δ) - (L : ℤ) ≤ j ∧ j ≤ round (a / δ) + (L : ℤ) := by
change j₀ - (L : ℤ) ≤ j ∧ j ≤ j₀ + (L : ℤ)
omega
apply Finset.mem_image.mpr
exact ⟨j, Finset.mem_Icc.mpr hmem, rfl⟩
end Falconer
end OAI
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