| import OAI.Geometry.PinnedDistances.Main |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open Filter |
| open scoped Topology BigOperators |
| namespace WeakPinned |
|
|
| def lightTargets (P : Finset Plane) (s : ℝ) (x : Plane) : Finset Plane := |
| (P.erase x).filter (fun y => (k P x y : ℝ) < (P.card : ℝ)^s) |
|
|
| def richTargets (P : Finset Plane) (s : ℝ) (x : Plane) : Finset Plane := |
| (P.erase x).filter (fun y => (P.card : ℝ)^s ≤ (k P x y : ℝ)) |
|
|
| lemma light_card_le (P : Finset Plane) (s : ℝ) (x : Plane) : |
| ((lightTargets P s x).card : ℝ) ≤ (distances P x).card * (P.card : ℝ)^s := by |
| classical |
| have hmap : (↑(lightTargets P s x) : Set Plane).MapsTo (fun y => dist y x) (distances P x) := by |
| intro y hy |
| exact Finset.mem_image_of_mem _ (Finset.mem_filter.mp hy).1 |
| have hcard := Finset.card_eq_sum_card_fiberwise hmap |
| rw [hcard, Nat.cast_sum] |
| calc |
| _ ≤ ∑ r ∈ distances P x, (P.card : ℝ)^s := by |
| apply Finset.sum_le_sum |
| intro r _ |
| let D := (lightTargets P s x).filter (fun y => dist y x = r) |
| by_cases hD : D.Nonempty |
| · obtain ⟨y,hy⟩ := hD |
| have hsub : D ⊆ distanceFiber P x y := by |
| intro z hz |
| obtain ⟨hzL,hzr⟩ := Finset.mem_filter.mp hz |
| obtain ⟨hyL,hyr⟩ := Finset.mem_filter.mp hy |
| exact Finset.mem_filter.mpr ⟨(Finset.mem_filter.mp hzL).1, hzr.trans hyr.symm⟩ |
| have hylt : (k P x y : ℝ) < (P.card : ℝ)^s := |
| (Finset.mem_filter.mp (Finset.mem_filter.mp hy).1).2 |
| exact (show (D.card : ℝ) ≤ (k P x y : ℝ) by exact_mod_cast Finset.card_le_card hsub).trans hylt.le |
| · change (D.card : ℝ) ≤ (P.card : ℝ)^s |
| rw [Finset.not_nonempty_iff_eq_empty.mp hD] |
| simp only [Finset.card_empty, Nat.cast_zero] |
| positivity |
| _ = _ := by simp |
|
|
| lemma row_card_split (P : Finset Plane) (s : ℝ) (x : Plane) : |
| (lightTargets P s x).card + (richTargets P s x).card = (P.erase x).card := by |
| classical |
| convert Finset.card_filter_add_card_filter_not (s := P.erase x) |
| (fun y => (k P x y : ℝ) < (P.card : ℝ)^s) using 1 |
| simp [lightTargets,richTargets] |
|
|
| lemma rich_card_sum (P : Finset Plane) (s : ℝ) : |
| ∑ x ∈ P, (richTargets P s x).card = (richPairs P s).card := by |
| classical |
| have hr (x : Plane) : richTargets P s x = P.filter (fun y => x ≠ y ∧ (P.card : ℝ)^s ≤ (k P x y : ℝ)) := by |
| ext y |
| simp only [richTargets, Finset.mem_filter, Finset.mem_erase] |
| tauto |
| simp only [hr, richPairs, Finset.card_eq_sum_ones, Finset.sum_filter, Finset.sum_product] |
|
|
| lemma bad_fraction_bound (P : Finset Plane) (ε : ℝ) (hn : 2 ≤ P.card) : |
| badPinFraction P ε ≤ 2*pairFraction P (ε/2) + |
| 2*((P.card : ℝ)^(1-ε/2)/(P.card : ℝ)) := by |
| classical |
| have hn2 : (2 : ℝ) ≤ P.card := by exact_mod_cast hn |
| have hn0 : (0 : ℝ) < P.card := by linarith |
| have hm0 : (0 : ℝ) < (P.card : ℝ)-1 := by linarith |
| have h1 : 1 ≤ P.card := by omega |
| have hlight (x) (hx : x ∈ badPins P ε) : |
| ((lightTargets P (ε/2) x).card : ℝ) ≤ (P.card : ℝ)^(1-ε/2) := by |
| have hd := (Finset.mem_filter.mp hx).2 |
| calc |
| _ ≤ (distances P x).card * (P.card : ℝ)^(ε/2) := light_card_le _ _ _ |
| _ ≤ (P.card : ℝ)^(1-ε)*(P.card : ℝ)^(ε/2) := |
| mul_le_mul_of_nonneg_right hd.le (Real.rpow_nonneg (by positivity) _) |
| _ = _ := by rw [← Real.rpow_add hn0]; congr 1; ring |
| have hrow (x) (hx : x ∈ badPins P ε) : |
| (P.card : ℝ)-1 ≤ (richTargets P (ε/2) x).card+(P.card : ℝ)^(1-ε/2) := by |
| have hxP : x ∈ P := (Finset.mem_filter.mp hx).1 |
| have he : ((lightTargets P (ε/2) x).card : ℝ)+(richTargets P (ε/2) x).card=(P.card : ℝ)-1 := by |
| rw [← Nat.cast_add, row_card_split, Finset.card_erase_of_mem hxP, Nat.cast_sub h1] |
| norm_num |
| linarith [hlight x hx] |
| have hsum : ((badPins P ε).card : ℝ)*((P.card : ℝ)-1) ≤ |
| (∑ x ∈ badPins P ε, ((richTargets P (ε/2) x).card : ℝ)) + |
| (badPins P ε).card*(P.card : ℝ)^(1-ε/2) := by |
| simpa only [Finset.sum_add_distrib, Finset.sum_const, nsmul_eq_mul] using |
| Finset.sum_le_sum hrow |
| have hRsum : (∑ x ∈ badPins P ε, ((richTargets P (ε/2) x).card : ℝ)) ≤ |
| ((richPairs P (ε/2)).card : ℝ) := by |
| calc |
| _ ≤ ∑ x ∈ P, ((richTargets P (ε/2) x).card : ℝ) := |
| Finset.sum_le_sum_of_subset_of_nonneg (Finset.filter_subset _ _) (by intros; positivity) |
| _ = _ := by rw [← Nat.cast_sum, rich_card_sum] |
| have hb : ((badPins P ε).card : ℝ) ≤ P.card := by |
| exact_mod_cast Finset.card_le_card (Finset.filter_subset (fun x => ((distances P x).card : ℝ)<(P.card : ℝ)^(1-ε)) P) |
| have hl0 : (0 : ℝ) ≤ (P.card : ℝ)^(1-ε/2) := Real.rpow_nonneg (by positivity) _ |
| have htotal : ((badPins P ε).card : ℝ)*((P.card : ℝ)-1) ≤ |
| (richPairs P (ε/2)).card+(P.card : ℝ)*(P.card : ℝ)^(1-ε/2) := by |
| linarith [mul_le_mul_of_nonneg_right hb hl0] |
| have hlratio : (P.card : ℝ)^(1-ε/2)/((P.card : ℝ)-1) ≤ |
| 2*((P.card : ℝ)^(1-ε/2)/(P.card : ℝ)) := by |
| rw [← mul_div_assoc, div_le_div_iff₀ hm0 hn0] |
| nlinarith |
| calc |
| badPinFraction P ε = ((badPins P ε).card : ℝ)*((P.card : ℝ)-1)/ |
| ((P.card : ℝ)*((P.card : ℝ)-1)) := by unfold badPinFraction; field_simp |
| _ ≤ ((richPairs P (ε/2)).card+(P.card : ℝ)*(P.card : ℝ)^(1-ε/2))/ |
| ((P.card : ℝ)*((P.card : ℝ)-1)) := |
| div_le_div_of_nonneg_right htotal (mul_pos hn0 hm0).le |
| _ = pairFraction P (ε/2)+(P.card : ℝ)^(1-ε/2)/((P.card : ℝ)-1) := by |
| unfold pairFraction |
| field_simp |
| _ ≤ 2*pairFraction P (ε/2)+2*((P.card : ℝ)^(1-ε/2)/(P.card : ℝ)) := by |
| linarith [pairFraction_nonneg P (ε/2)] |
|
|
| lemma badPinFraction_nonneg (P : Finset Plane) (ε : ℝ) : 0 ≤ badPinFraction P ε := by |
| unfold badPinFraction |
| positivity |
|
|
| lemma badPinFraction_le_one (P : Finset Plane) (ε : ℝ) : badPinFraction P ε ≤ 1 := by |
| have hb : ((badPins P ε).card : ℝ) ≤ P.card := by |
| exact_mod_cast Finset.card_le_card (Finset.filter_subset (fun x => ((distances P x).card : ℝ)<(P.card : ℝ)^(1-ε)) P) |
| by_cases hn : P.card=0 |
| · simp [badPinFraction,hn] |
| · rw [badPinFraction,div_le_one (by exact_mod_cast Nat.pos_of_ne_zero hn)] |
| exact hb |
|
|
| lemma bad_values_nonempty (n : ℕ) (ε : ℝ) : |
| ({a : ℝ | ∃ P : Finset Plane, P.card=n ∧ a=badPinFraction P ε}).Nonempty := by |
| obtain ⟨P,hP⟩ := Finset.exists_card_eq (α:=Plane) n |
| exact ⟨_,P,hP,rfl⟩ |
|
|
| lemma bad_values_bddAbove (n : ℕ) (ε : ℝ) : |
| BddAbove {a : ℝ | ∃ P : Finset Plane, P.card=n ∧ a=badPinFraction P ε} := by |
| refine ⟨1,?_⟩ |
| rintro a ⟨P,_,rfl⟩ |
| exact badPinFraction_le_one P ε |
|
|
| lemma badPinFraction_le_B (P : Finset Plane) (ε : ℝ) (hn : 0<P.card) : |
| badPinFraction P ε ≤ B P.card ε := by |
| rw [B,ite_eq_left hn] |
| exact le_csSup (bad_values_bddAbove _ _) ⟨P,rfl,rfl⟩ |
|
|
| lemma B_nonneg (n : ℕ) (ε : ℝ) : 0 ≤ B n ε := by |
| by_cases hn : 0<n |
| · obtain ⟨P,hP⟩ := Finset.exists_card_eq (α:=Plane) n |
| rw [← hP] at hn ⊢ |
| exact (badPinFraction_nonneg P ε).trans (badPinFraction_le_B P ε hn) |
| · simp [B,hn] |
|
|
| lemma B_bound (n : ℕ) (hn : 2≤n) (ε : ℝ) : |
| B n ε ≤ 2*F n (ε/2)+2*((n : ℝ)^(1-ε/2)/(n : ℝ)) := by |
| rw [B,ite_eq_left (by omega : 0<n)] |
| apply csSup_le (bad_values_nonempty _ _) |
| rintro a ⟨P,hP,rfl⟩ |
| have hPn : 2≤P.card := by omega |
| have hf := fraction_le_F P (ε/2) hPn |
| have hb := bad_fraction_bound P ε hPn |
| rw [hP] at hf hb |
| linarith |
|
|
| /-- All but a uniformly vanishing fraction of pins determine at least |
| `n^(1-ε)` distinct nonzero distances. -/ |
| theorem pins (ε : ℝ) (hε : 0<ε) : Tendsto (fun n : ℕ => B n ε) atTop (𝓝 0) := by |
| have hpower : Tendsto (fun n : ℕ => (n : ℝ)^(1-ε/2)/(n : ℝ)) atTop (𝓝 0) := by |
| have ht := (tendsto_rpow_neg_atTop (half_pos hε)).comp |
| (tendsto_natCast_atTop_atTop (R:=ℝ)) |
| apply ht.congr' |
| filter_upwards [eventually_ge_atTop 1] with n hn |
| have hn0 : (0 : ℝ)<n := by exact_mod_cast (show 0<n by omega) |
| simp only [Function.comp_apply] |
| rw [show 1-ε/2=1+(-(ε/2)) by ring, Real.rpow_add hn0, Real.rpow_one] |
| exact (mul_div_cancel_left₀ _ hn0.ne').symm |
| apply squeeze_zero' (Eventually.of_forall (fun n => B_nonneg n ε)) |
| (eventually_atTop.2 ⟨2,fun n hn => B_bound n hn ε⟩) |
| simpa only [mul_zero,add_zero] using ((main (ε/2) (half_pos hε)).const_mul 2).add (hpower.const_mul 2) |
| |
| theorem pinsStatement : PinsStatement := pins |
| |
| end WeakPinned |
| |
| end |
| end |
| section |
| noncomputable section |
| open Filter |
| open scoped Topology |
| namespace WeakPinned |
| |
| lemma distances_pos {P : Finset Plane} {x : Plane} {r : ℝ} (hr : r ∈ distances P x) : |
| 0 < r := by |
| obtain ⟨y,hy,rfl⟩ := Finset.mem_image.mp hr |
| exact dist_pos.mpr (Finset.mem_erase.mp hy).1 |
| |
| /-- Every sufficiently large configuration has a pin with at least |
| the prescribed power of its cardinality many distinct nonzero distances. -/ |
| theorem exists_pin_eventually (ε : ℝ) (hε : 0 < ε) : |
| ∀ᶠ n : ℕ in atTop, ∀ P : Finset Plane, P.card = n → |
| ∃ x ∈ P, (P.card : ℝ) ^ (1-ε) ≤ ((distances P x).card : ℝ) := by |
| have hb : ∀ᶠ n : ℕ in atTop, B n ε < 1 := |
| (pins ε hε).eventually (gt_mem_nhds (by norm_num : (0 : ℝ) < 1)) |
| filter_upwards [hb, eventually_ge_atTop 1] with n hn hn1 P hP |
| have hP0 : 0 < P.card := by omega |
| by_contra hf |
| have hbad : badPins P ε = P := by |
| apply Finset.filter_eq_self.mpr |
| intro x hx |
| exact lt_of_not_ge (fun h => hf ⟨x,hx,h⟩) |
| have hfrac : badPinFraction P ε = 1 := by |
| rw [badPinFraction, hbad, div_self (by exact_mod_cast Nat.ne_of_gt hP0)] |
| have he := badPinFraction_le_B P ε hP0 |
| rw [hfrac,hP] at he |
| exact (not_lt_of_ge he) hn |
| |
| end WeakPinned |
| |
| end |
| end |
| |
| end OAI |
| |
