| import OAI.MeasureTheory.Falconer.Incidence.RoundedRowPencils |
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| namespace OAI |
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| /-! Normalize the rounded product rows into the bounded line chart without |
| changing their first coordinate. All size losses are explicit. -/ |
| namespace Falconer |
| attribute [local instance] Classical.propDecidable |
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| noncomputable def verticalQuarter (p : Space 2) : Space 2 := rowPoint (p 0) (p 1/4) |
| noncomputable def verticalFour (p : Space 2) : Space 2 := rowPoint (p 0) (4*p 1) |
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| @[simp] theorem verticalQuarter_zero (p : Space 2) : verticalQuarter p 0 = p 0 := rfl |
| @[simp] theorem verticalQuarter_one (p : Space 2) : verticalQuarter p 1 = p 1/4 := rfl |
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| theorem verticalFour_verticalQuarter (p : Space 2) : verticalFour (verticalQuarter p) = p := by |
| ext i |
| fin_cases i |
| · rfl |
| · change 4*(p 1/4) = p 1 |
| ring |
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|
| theorem verticalQuarter_verticalFour (p : Space 2) : verticalQuarter (verticalFour p) = p := by |
| ext i |
| fin_cases i |
| · rfl |
| · change (4*p 1)/4 = p 1 |
| ring |
|
|
| theorem verticalQuarter_injective : Function.Injective verticalQuarter := |
| Function.LeftInverse.injective verticalFour_verticalQuarter |
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| theorem verticalQuarter_row_distance {p q : Space 2} (h : p 0 = q 0) : |
| dist (verticalQuarter p) (verticalQuarter q) = dist p q/4 := by |
| have hp : p = rowPoint (p 0) (p 1) := by ext i; fin_cases i <;> rfl |
| have hq : q = rowPoint (p 0) (q 1) := by |
| ext i |
| fin_cases i |
| · exact h.symm |
| · rfl |
| conv_rhs => rw [hp,hq,rowPoint_dist] |
| change dist (rowPoint (p 0) (p 1/4)) (rowPoint (q 0) (q 1/4)) = _ |
| rw [← h,rowPoint_dist,← sub_div,abs_div] |
| norm_num |
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| theorem verticalQuarter_row_separated {δ y : ℝ} (P : Finset (Space 2)) |
| (hrow : ∀ p ∈ P, p 0 = y) (hsep : FiniteSeparatedAt δ P) : |
| FiniteSeparatedAt (δ/4) (P.image verticalQuarter) := by |
| intro p hp q hq hpq |
| obtain ⟨x,hx,rfl⟩ := Finset.mem_image.mp hp |
| obtain ⟨z,hz,rfl⟩ := Finset.mem_image.mp hq |
| rw [verticalQuarter_row_distance ((hrow x hx).trans (hrow z hz).symm)] |
| exact div_le_div_of_nonneg_right |
| (hsep x hx z hz (fun h => hpq (congrArg _ h))) (by norm_num) |
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| theorem verticalQuarter_row_regular {δ s K y : ℝ} (hδ : 0 < δ) |
| (hs : 0 ≤ s) (hK : 1 ≤ K) (P : Finset (Space 2)) |
| (hrow : ∀ p ∈ P, p 0 = y) (hreg : NormalizedFiniteBallBound δ s K P) : |
| NormalizedFiniteBallBound (δ/4) s (K*(4:ℝ)^s) (P.image verticalQuarter) := by |
| intro z r hr _ |
| have hr0 : 0 < r := (by linarith : 0 < δ/4).trans_le hr |
| have hsub : (P.image verticalQuarter).filter (fun p => dist p z ≤ r) ⊆ |
| (P.filter (fun p => dist p (rowPoint y (4*z 1)) ≤ 4*r)).image verticalQuarter := by |
| intro p hp |
| obtain ⟨hp,hpr⟩ := Finset.mem_filter.mp hp |
| obtain ⟨x,hx,rfl⟩ := Finset.mem_image.mp hp |
| refine Finset.mem_image.mpr ⟨x,Finset.mem_filter.mpr ⟨hx,?_⟩,rfl⟩ |
| rw [row_dist_projection (hrow x hx)] |
| have hh : |x 1/4-z 1| ≤ r := |
| (show |x 1/4-z 1| ≤ dist (verticalQuarter x) z by |
| simpa only [Real.dist_eq,verticalQuarter_one] using PiLp.dist_apply_le (verticalQuarter x) z 1).trans hpr |
| have he : |x 1-4*z 1| = 4*|x 1/4-z 1| := by |
| rw [← abs_of_pos (by norm_num : (0:ℝ) < 4),← abs_mul] |
| congr 1 |
| ring |
| rw [he] |
| linarith only [hh] |
| have hc : (((P.image verticalQuarter).filter (fun p => dist p z ≤ r)).card:ℝ) ≤ |
| ((P.filter (fun p => dist p (rowPoint y (4*z 1)) ≤ 4*r)).card:ℝ) := by |
| exact_mod_cast (Finset.card_le_card hsub).trans Finset.card_image_le |
| rw [Finset.card_image_of_injective _ verticalQuarter_injective] |
| calc |
| _ ≤ ((P.filter (fun p => dist p (rowPoint y (4*z 1)) ≤ 4*r)).card:ℝ) := hc |
| _ ≤ K*(4*r)^s*(P.card:ℝ) := |
| normalized_ball_bound_all_radii hs hK hreg _ _ (by linarith only [hr]) |
| _ = _ := by rw [Real.mul_rpow (by norm_num : (0:ℝ) ≤ 4) hr0.le]; ring |
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| theorem quarter_adjusted_incident {δ : ℝ} (hδ : 0 < δ) (p l : Space 2) : |
| TubeIncident (δ/4) (verticalQuarter p) (planarQuarter (rowAdjustedLine δ p l)) := by |
| have hd : 1 ≤ Real.sqrt (1+(planarQuarter (rowAdjustedLine δ p l) 0)^2) := by |
| apply (Real.le_sqrt (by norm_num) (by positivity)).2 |
| nlinarith only [sq_nonneg (planarQuarter (rowAdjustedLine δ p l) 0)] |
| have he : verticalQuarter p 1-planarQuarter (rowAdjustedLine δ p l) 0*verticalQuarter p 0- |
| planarQuarter (rowAdjustedLine δ p l) 1 = |
| (p 1-rowAdjustedLine δ p l 0*p 0-rowAdjustedLine δ p l 1)/4 := by |
| simp only [verticalQuarter_zero,verticalQuarter_one,planarQuarter_apply] |
| ring |
| unfold TubeIncident |
| rw [he,abs_div] |
| norm_num only [abs_of_pos (by norm_num : (0:ℝ) < 4)] |
| apply (div_le_iff₀ (zero_lt_one.trans_le hd)).2 |
| exact (div_le_div_of_nonneg_right (rowAdjustedLine_residual hδ p l) (by norm_num)).trans |
| (le_mul_of_one_le_right (by positivity) hd) |
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| theorem quarter_adjusted_chart {δ : ℝ} (hδ : 0 < δ) (hδsmall : δ ≤ 1/8) |
| (p l : Space 2) (hp : |p 0| ≤ 1) (hl : |l 0| ≤ 1 ∧ |l 1| ≤ 1) |
| (hi : TubeIncident (3*δ) p l) : |
| |planarQuarter (rowAdjustedLine δ p l) 0| ≤ 1 ∧ |
| |planarQuarter (rowAdjustedLine δ p l) 1| ≤ 1 := by |
| obtain ⟨h0,h1⟩ := rowAdjustedLine_coordinate_error hδ p l hp hl.1 hi |
| have h0' : |rowAdjustedLine δ p l 0| ≤ |l 0|+δ := by |
| have hh := abs_sub_le (rowAdjustedLine δ p l 0) (l 0) 0 |
| simp only [sub_zero] at hh |
| rw [abs_sub_comm (rowAdjustedLine δ p l 0) (l 0)] at hh |
| linarith only [hh,h0] |
| have h1' : |rowAdjustedLine δ p l 1| ≤ |l 1|+8*δ := by |
| have hh := abs_sub_le (rowAdjustedLine δ p l 1) (l 1) 0 |
| simp only [sub_zero] at hh |
| rw [abs_sub_comm (rowAdjustedLine δ p l 1) (l 1)] at hh |
| linarith only [hh,h1] |
| simp only [planarQuarter_apply,abs_div,abs_of_pos (by norm_num : (0:ℝ) < 4)] |
| constructor <;> linarith only [h0',h1',hl.1,hl.2,hδsmall] |
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| end Falconer |
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| end OAI |
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