| import OAI.NumberTheory.Jacobsthal.Estimates.RegularStripBound | |
| import OAI.NumberTheory.Jacobsthal.Estimates.SortedRealGaps | |
| namespace OAI | |
| namespace Erdos970 | |
| section | |
| open Set | |
| namespace ErdosMarkedStrips | |
| open ErdosImplicitCurvature ErdosLocalFiberGraphs ErdosRegularStrip ErdosConvexGraph | |
| attribute [local instance] Classical.propDecidable | |
| noncomputable def polynomialSquarePoints (Q : Bivariate) (S : ℝ) : Finset (ℤ × ℤ) := | |
| ((Finset.Icc (0 : ℤ) ⌊S⌋).product (Finset.Icc (0 : ℤ) ⌊S⌋)).filter | |
| (fun p => peval Q p.1 p.2 = 0) | |
| theorem mem_polynomialSquarePoints (Q : Bivariate) (S : ℝ) (p : ℤ × ℤ) : | |
| p ∈ polynomialSquarePoints Q S ↔ InSquare S p ∧ peval Q p.1 p.2 = 0 := by | |
| simp only [polynomialSquarePoints,InSquare,Finset.mem_filter,Finset.product_eq_sprod, | |
| Finset.mem_product,Finset.mem_Icc,Int.le_floor,Int.cast_nonneg_iff] | |
| noncomputable def markedFiberPoints (Q : Bivariate) (t S : ℝ) : Finset (ℤ × ℤ) := | |
| (polynomialSquarePoints Q S).filter (fun p => (p.1 : ℝ) = t) | |
| theorem mem_markedFiberPoints (Q : Bivariate) (t S : ℝ) (p : ℤ × ℤ) : | |
| p ∈ markedFiberPoints Q t S ↔ p ∈ polynomialSquarePoints Q S ∧ (p.1 : ℝ) = t := by | |
| simp only [markedFiberPoints,Finset.mem_filter] | |
| theorem markedFiber_card_le (Q : Bivariate) (t S : ℝ) | |
| (hp : verticalPolynomial Q t ≠ 0) : (markedFiberPoints Q t S).card ≤ Q.totalDegree := by | |
| classical | |
| obtain ⟨z,hz⟩ : ∃ z : ℝ, peval Q t z ≠ 0 := by | |
| by_contra! h | |
| apply hp | |
| apply Polynomial.funext | |
| intro z | |
| rw [verticalPolynomial_eval,h z,Polynomial.eval_zero] | |
| have hfin := boundedFiber_finite Q t 0 S z hz | |
| have hmap : ∀ p ∈ (markedFiberPoints Q t S : Set (ℤ × ℤ)), | |
| (p.2 : ℝ) ∈ boundedFiber Q t 0 S := by | |
| intro p hp | |
| have hm := (mem_markedFiberPoints Q t S p).mp hp | |
| have hs := (mem_polynomialSquarePoints Q S p).mp hm.1 | |
| refine ⟨hs.1.2,?_⟩ | |
| simpa only [hm.2] using hs.2 | |
| have hinj : InjOn (fun p : ℤ × ℤ => (p.2 : ℝ)) (markedFiberPoints Q t S : Set (ℤ × ℤ)) := by | |
| intro p hp q hq hpq | |
| have hp1 := ((mem_markedFiberPoints Q t S p).mp hp).2 | |
| have hq1 := ((mem_markedFiberPoints Q t S q).mp hq).2 | |
| apply Prod.ext | |
| · exact_mod_cast hp1.trans hq1.symm | |
| · change (p.2 : ℝ) = (q.2 : ℝ) at hpq | |
| exact_mod_cast hpq | |
| have hc := ncard_le_ncard_of_injOn (fun p : ℤ × ℤ => (p.2 : ℝ)) hmap hinj hfin | |
| simp only [ncard_coe_finset] at hc | |
| exact hc.trans (boundedFiber_card_le_degree Q t 0 S z hz) | |
| theorem square_points_partition (Q : Bivariate) (S : ℝ) (E : Finset ℝ) | |
| (h0 : 0 ∈ E) (hS : S ∈ E) : | |
| polynomialSquarePoints Q S = | |
| E.biUnion (fun t => markedFiberPoints Q t S) ∪ | |
| Finset.univ.biUnion (fun i : Fin (E.card-1) => | |
| polynomialStripPoints Q (gapLeft E i) (gapRight E i) S) := by | |
| classical | |
| ext p | |
| constructor | |
| · intro hp | |
| have hs := (mem_polynomialSquarePoints Q S p).mp hp | |
| by_cases hm : (p.1 : ℝ) ∈ E | |
| · apply Finset.mem_union_left | |
| exact Finset.mem_biUnion.mpr ⟨p.1,hm,(mem_markedFiberPoints Q p.1 S p).mpr ⟨hp,rfl⟩⟩ | |
| · obtain ⟨i,hi⟩ := exists_gap_of_not_mem E h0 hS hs.1.1 hm | |
| apply Finset.mem_union_right | |
| exact Finset.mem_biUnion.mpr ⟨i,Finset.mem_univ i, | |
| (mem_polynomialStripPoints Q _ _ S p).mpr ⟨hs.1,hi,hs.2⟩⟩ | |
| · intro hp | |
| rcases Finset.mem_union.mp hp with hp | hp | |
| · obtain ⟨t,_,ht⟩ := Finset.mem_biUnion.mp hp | |
| exact ((mem_markedFiberPoints Q t S p).mp ht).1 | |
| · obtain ⟨i,_,hi⟩ := Finset.mem_biUnion.mp hp | |
| have hs := (mem_polynomialStripPoints Q _ _ S p).mp hi | |
| exact (mem_polynomialSquarePoints Q S p).mpr ⟨hs.1,hs.2.2⟩ | |
| end ErdosMarkedStrips | |
| end | |
| section | |
| open Set | |
| namespace ErdosMarkedStrips | |
| open ErdosImplicitCurvature ErdosLocalFiberGraphs ErdosRegularStrip ErdosConvexGraph | |
| def RegularOff (Q : Bivariate) (S : ℝ) (E : Finset ℝ) : Prop := | |
| ∀ t ∉ E, (peval Q t 0 ≠ 0 ∧ peval Q t S ≠ 0) ∧ | |
| ∀ y ∈ Icc 0 S, peval Q t y = 0 → | |
| peval (partialX Q) t y ≠ 0 ∧ peval (partialY Q) t y ≠ 0 ∧ | |
| peval (inflectionPolynomial Q) t y ≠ 0 | |
| theorem marked_square_lattice_bound (Q : Bivariate) (S : ℝ) (E : Finset ℝ) | |
| (hS : 0 ≤ S) (h0 : 0 ∈ E) (hSE : S ∈ E) | |
| (hfiber : ∀ t ∈ E, verticalPolynomial Q t ≠ 0) (hregular : RegularOff Q S E) : | |
| ((polynomialSquarePoints Q S).card : ℝ) ≤ | |
| 13*(E.card : ℝ)*(Q.totalDegree : ℝ)*(1+S^((2 : ℝ)/3)) := by | |
| classical | |
| let B := 1+S^((2 : ℝ)/3) | |
| have hB : 1 ≤ B := by | |
| dsimp [B] | |
| exact le_add_of_nonneg_right (Real.rpow_nonneg hS _) | |
| have hB0 : 0 ≤ B := le_trans zero_le_one hB | |
| have hstrip (i : Fin (E.card-1)) : | |
| ((polynomialStripPoints Q (gapLeft E i) (gapRight E i) S).card : ℝ) ≤ | |
| 12*(Q.totalDegree : ℝ)*B := by | |
| apply regular_strip_lattice_bound Q _ _ S hS | |
| · exact fun t ht => (hregular t (no_mark_in_gap E i ht)).1 | |
| · exact fun t ht => (hregular t (no_mark_in_gap E i ht)).2 | |
| have hcount : (polynomialSquarePoints Q S).card ≤ | |
| (∑ t ∈ E, (markedFiberPoints Q t S).card) + | |
| ∑ i : Fin (E.card-1), (polynomialStripPoints Q (gapLeft E i) (gapRight E i) S).card := by | |
| rw [square_points_partition Q S E h0 hSE] | |
| exact (Finset.card_union_le _ _).trans | |
| (Nat.add_le_add Finset.card_biUnion_le Finset.card_biUnion_le) | |
| have hm : (∑ t ∈ E, ((markedFiberPoints Q t S).card : ℝ)) ≤ | |
| (E.card : ℝ)*(Q.totalDegree : ℝ) := by | |
| calc | |
| _ ≤ ∑ _t ∈ E, (Q.totalDegree : ℝ) := Finset.sum_le_sum (fun t ht => by | |
| exact_mod_cast markedFiber_card_le Q t S (hfiber t ht)) | |
| _ = _ := by simp | |
| have hg : (∑ i : Fin (E.card-1), | |
| ((polynomialStripPoints Q (gapLeft E i) (gapRight E i) S).card : ℝ)) ≤ | |
| (E.card : ℝ)*(12*(Q.totalDegree : ℝ)*B) := by | |
| calc | |
| _ ≤ ∑ _i : Fin (E.card-1), 12*(Q.totalDegree : ℝ)*B := | |
| Finset.sum_le_sum (fun i _ => hstrip i) | |
| _ = ((E.card-1 : ℕ) : ℝ)*(12*(Q.totalDegree : ℝ)*B) := by simp | |
| _ ≤ _ := mul_le_mul_of_nonneg_right (by exact_mod_cast Nat.sub_le E.card 1) (by positivity) | |
| have hc : ((polynomialSquarePoints Q S).card : ℝ) ≤ | |
| (∑ t ∈ E, ((markedFiberPoints Q t S).card : ℝ)) + | |
| ∑ i : Fin (E.card-1), ((polynomialStripPoints Q (gapLeft E i) (gapRight E i) S).card : ℝ) := by | |
| exact_mod_cast hcount | |
| calc | |
| ((polynomialSquarePoints Q S).card : ℝ) ≤ | |
| (E.card : ℝ)*(Q.totalDegree : ℝ) + (E.card : ℝ)*(12*(Q.totalDegree : ℝ)*B) := | |
| hc.trans (add_le_add hm hg) | |
| _ ≤ (E.card : ℝ)*(Q.totalDegree : ℝ)*B + (E.card : ℝ)*(12*(Q.totalDegree : ℝ)*B) := by | |
| have hscale : (E.card : ℝ)*(Q.totalDegree : ℝ) ≤ | |
| (E.card : ℝ)*(Q.totalDegree : ℝ)*B := by | |
| calc | |
| _ = (E.card : ℝ)*(Q.totalDegree : ℝ)*1 := by ring | |
| _ ≤ _ := mul_le_mul_of_nonneg_left hB (by positivity) | |
| exact add_le_add hscale le_rfl | |
| _ = 13*(E.card : ℝ)*(Q.totalDegree : ℝ)*(1+S^((2 : ℝ)/3)) := by dsimp [B]; ring | |
| theorem marked_square_cubic_bound (Q : Bivariate) (S : ℝ) (E : Finset ℝ) (C : ℕ) | |
| (hS : 0 ≤ S) (h0 : 0 ∈ E) (hSE : S ∈ E) | |
| (hfiber : ∀ t ∈ E, verticalPolynomial Q t ≠ 0) (hregular : RegularOff Q S E) | |
| (hcard : E.card ≤ C*Q.totalDegree^2) : | |
| ((polynomialSquarePoints Q S).card : ℝ) ≤ | |
| 13*(C : ℝ)*(Q.totalDegree : ℝ)^3*(1+S^((2 : ℝ)/3)) := by | |
| calc | |
| ((polynomialSquarePoints Q S).card : ℝ) ≤ | |
| 13*(E.card : ℝ)*(Q.totalDegree : ℝ)*(1+S^((2 : ℝ)/3)) := | |
| marked_square_lattice_bound Q S E hS h0 hSE hfiber hregular | |
| _ ≤ 13*((C : ℝ)*(Q.totalDegree : ℝ)^2)*(Q.totalDegree : ℝ)*(1+S^((2 : ℝ)/3)) := by | |
| gcongr | |
| exact_mod_cast hcard | |
| _ = _ := by ring | |
| end ErdosMarkedStrips | |
| end | |
| end Erdos970 | |
| end OAI | |