import Mathlib import OAI.GroupTheory.SimpleAmenable.Arithmetic.PolygonArithmetic namespace OAI section section open scoped symmDiff namespace SimpleAmenable section LatticeCounting def embeddingBox (x y a b : ℝ) : Set CutRing := {z | ordinary z ∈ Set.Icc x (x + a) ∧ conjugate z ∈ Set.Icc y (y + b)} theorem embeddingBox_floor_injective (x y a b : ℝ) (hb : 0 < b) : Set.InjOn (fun z : CutRing => ⌊b * ordinary z⌋) (embeddingBox x y a b) := by intro z hz w hw he dsimp only at he by_contra hn have hnorm := one_le_abs_cut_norm (sub_ne_zero.mpr hn) rw [map_sub, map_sub, abs_mul] at hnorm have ho : |b * (ordinary z - ordinary w)| < 1 := by have hz₁ := Int.floor_le (b * ordinary z) have hz₂ := Int.lt_floor_add_one (b * ordinary z) have hw₁ := Int.floor_le (b * ordinary w) have hw₂ := Int.lt_floor_add_one (b * ordinary w) rw [he] at hz₁ hz₂ apply abs_lt.mpr constructor <;> nlinarith have hc : |conjugate z - conjugate w| ≤ b := by exact abs_le.mpr ⟨by linarith [hz.2.1, hw.2.2], by linarith [hz.2.2, hw.2.1]⟩ rw [abs_mul, abs_of_pos hb] at ho have hprod := mul_le_mul_of_nonneg_left hc (abs_nonneg (ordinary z - ordinary w)) nlinarith theorem embeddingBox_floor_mapsTo (x y a b : ℝ) (hb : 0 ≤ b) : Set.MapsTo (fun z : CutRing => ⌊b * ordinary z⌋) (embeddingBox x y a b) (Set.Icc ⌊b * x⌋ ⌊b * (x + a)⌋) := by intro z hz exact ⟨Int.floor_mono (mul_le_mul_of_nonneg_left hz.1.1 hb), Int.floor_mono (mul_le_mul_of_nonneg_left hz.1.2 hb)⟩ theorem embeddingBox_finite (x y a b : ℝ) (hb : 0 < b) : (embeddingBox x y a b).Finite := (Set.finite_Icc _ _).of_injOn (embeddingBox_floor_mapsTo x y a b hb.le) (embeddingBox_floor_injective x y a b hb) theorem embeddingBox_card_le (x y a b : ℝ) (ha : 0 ≤ a) (hb : 0 < b) (s : Finset CutRing) (hs : (s : Set CutRing) ⊆ embeddingBox x y a b) : (s.card : ℝ) ≤ a * b + 2 := by classical have hinj := (embeddingBox_floor_injective x y a b hb).mono hs have hcard : s.card ≤ (Finset.Icc ⌊b * x⌋ ⌊b * (x + a)⌋).card := by calc _ = (s.image (fun z => ⌊b * ordinary z⌋)).card := (Finset.card_image_of_injOn hinj).symm _ ≤ _ := Finset.card_le_card (by intro k hk obtain ⟨z, hz, rfl⟩ := Finset.mem_image.mp hk exact Finset.mem_Icc.mpr (embeddingBox_floor_mapsTo x y a b hb.le (hs hz))) have hle : ⌊b * x⌋ ≤ ⌊b * (x + a)⌋ + 1 := by have := Int.floor_mono (show b * x ≤ b * (x + a) by nlinarith) omega have he := Int.card_Icc_of_le ⌊b * x⌋ ⌊b * (x + a)⌋ hle have he' : ((Finset.Icc ⌊b * x⌋ ⌊b * (x + a)⌋).card : ℝ) = (⌊b * (x + a)⌋ : ℝ) + 1 - (⌊b * x⌋ : ℝ) := by exact_mod_cast he have hc : (s.card : ℝ) ≤ ((Finset.Icc ⌊b * x⌋ ⌊b * (x + a)⌋).card : ℝ) := by exact_mod_cast hcard rw [he'] at hc linarith [Int.floor_le (b * (x + a)), Int.lt_floor_add_one (b * x)] def dividedEmbeddingBox (d x y a b : ℝ) : Set CutRing := {z | ordinary z / d ∈ Set.Icc x (x + a) ∧ conjugate z / d ∈ Set.Icc y (y + b)} theorem dividedEmbeddingBox_eq (d x y a b : ℝ) (hd : 0 < d) : dividedEmbeddingBox d x y a b = embeddingBox (d*x) (d*y) (d*a) (d*b) := by ext z simp only [dividedEmbeddingBox, embeddingBox, Set.mem_ofPred_eq, Set.mem_Icc, le_div_iff₀ hd, div_le_iff₀ hd] constructor <;> rintro ⟨⟨h₁,h₂⟩,⟨h₃,h₄⟩⟩ <;> constructor <;> constructor <;> nlinarith theorem dividedEmbeddingBox_finite (d x y a b : ℝ) (hd : 0 < d) (hb : 0 < b) : (dividedEmbeddingBox d x y a b).Finite := by rw [dividedEmbeddingBox_eq d x y a b hd] exact embeddingBox_finite _ _ _ _ (mul_pos hd hb) theorem dividedEmbeddingBox_card_le (d x y a b : ℝ) (hd : 0 < d) (ha : 0 ≤ a) (hb : 0 < b) (s : Finset CutRing) (hs : (s : Set CutRing) ⊆ dividedEmbeddingBox d x y a b) : (s.card : ℝ) ≤ d^2 * a * b + 2 := by rw [dividedEmbeddingBox_eq d x y a b hd] at hs convert embeddingBox_card_le _ _ _ _ (mul_nonneg hd.le ha) (mul_pos hd hb) s hs using 1; ring theorem pairedEmbeddingBox_card_le (d : ℝ) (hd : 0 < d) (x₁ y₁ a₁ b₁ x₂ y₂ a₂ b₂ : ℝ) (ha₁ : 0 ≤ a₁) (hb₁ : 0 < b₁) (ha₂ : 0 ≤ a₂) (hb₂ : 0 < b₂) (s : Finset (CutRing × CutRing)) (hs : ∀ z ∈ s, z.1 ∈ dividedEmbeddingBox d x₁ y₁ a₁ b₁ ∧ z.2 ∈ dividedEmbeddingBox d x₂ y₂ a₂ b₂) : (s.card : ℝ) ≤ (d^2*a₁*b₁ + 2) * (d^2*a₂*b₂ + 2) := by classical have hc₁ := dividedEmbeddingBox_card_le d x₁ y₁ a₁ b₁ hd ha₁ hb₁ (s.image Prod.fst) (by rintro z hz obtain ⟨p,hp,rfl⟩ := Finset.mem_image.mp hz exact (hs p hp).1) have hc₂ := dividedEmbeddingBox_card_le d x₂ y₂ a₂ b₂ hd ha₂ hb₂ (s.image Prod.snd) (by rintro z hz obtain ⟨p,hp,rfl⟩ := Finset.mem_image.mp hz exact (hs p hp).2) have hc : s.card ≤ (s.image Prod.fst).card * (s.image Prod.snd).card := by rw [← Finset.card_product] apply Finset.card_le_card intro z hz exact Finset.mem_product.mpr ⟨Finset.mem_image_of_mem _ hz, Finset.mem_image_of_mem _ hz⟩ calc (s.card : ℝ) ≤ (s.image Prod.fst).card * (s.image Prod.snd).card := by exact_mod_cast hc _ ≤ _ := mul_le_mul hc₁ hc₂ (by positivity) (by positivity) end LatticeCounting section QuantitativeArithmetic lemma floor_error (x : ℝ) : |(⌊x⌋ : ℝ) - x| ≤ 1 := by rw [abs_of_nonpos (sub_nonpos.mpr (Int.floor_le x))] linarith [Int.lt_floor_add_one x] theorem quadratic_lattice_cover (u : CutRing) (hu : u ≠ 0) (x y : ℝ) : ∃ z : CutRing, |ordinary z - x| ≤ 3 * |ordinary u| ∧ |conjugate z - y| ≤ 2 * |conjugate u| := by have huo : ordinary u ≠ 0 := by simpa using ordinary_injective.ne hu have huc : conjugate u ≠ 0 := by simpa using conjugate_injective.ne hu have hd : Real.goldenRatio - Real.goldenConj ≠ 0 := by rw [Real.goldenRatio_sub_goldenConj] exact ne_of_gt (Real.sqrt_pos.mpr (by norm_num)) let b := (x / ordinary u - y / conjugate u) / (Real.goldenRatio - Real.goldenConj) let a := x / ordinary u - b * Real.goldenRatio have hb : b * (Real.goldenRatio - Real.goldenConj) = x / ordinary u - y / conjugate u := div_mul_cancel₀ _ hd have ha₁ : a + b * Real.goldenRatio = x / ordinary u := by dsimp [a]; ring have ha₂ : a + b * Real.goldenConj = y / conjugate u := by dsimp [a]; nlinarith [hb] have hx : ordinary u * (a + b * Real.goldenRatio) = x := by rw [ha₁]; exact mul_div_cancel₀ _ huo have hy : conjugate u * (a + b * Real.goldenConj) = y := by rw [ha₂]; exact mul_div_cancel₀ _ huc let z : CutRing := u * ((⌊a⌋ : CutRing) + (⌊b⌋ : CutRing) * cutTau) have ho : ordinary z - x = ordinary u * (((⌊a⌋ : ℝ) - a) + ((⌊b⌋ : ℝ) - b) * Real.goldenRatio) := by dsimp [z] simp only [map_mul,map_add,map_intCast,ordinary_cutTau] rw [← hx]; ring have hc : conjugate z - y = conjugate u * (((⌊a⌋ : ℝ) - a) + ((⌊b⌋ : ℝ) - b) * Real.goldenConj) := by dsimp [z] simp only [map_mul,map_add,map_intCast,conjugate_cutTau] rw [← hy]; ring have hr : |Real.goldenRatio| ≤ 2 := by rw [abs_of_pos Real.goldenRatio_pos] exact Real.goldenRatio_lt_two.le have hb' : |Real.goldenConj| ≤ 1 := by rw [abs_of_neg Real.goldenConj_neg] linarith [Real.neg_one_lt_goldenConj] have bound (t : ℝ) (B : ℝ) (ht : |t| ≤ B) : |((⌊a⌋ : ℝ) - a) + ((⌊b⌋ : ℝ) - b)*t| ≤ 1+B := by calc _ ≤ |(⌊a⌋ : ℝ)-a| + |((⌊b⌋ : ℝ)-b)*t| := abs_add_le _ _ _ = |(⌊a⌋ : ℝ)-a| + |(⌊b⌋ : ℝ)-b| *|t| := by rw [abs_mul] _ ≤ 1 + 1*B := by gcongr; exact floor_error a; exact floor_error b _ = _ := by ring refine ⟨z,?_,?_⟩ · rw [ho,abs_mul] have := mul_le_mul_of_nonneg_left (bound _ _ hr) (abs_nonneg (ordinary u)) nlinarith · rw [hc,abs_mul] have := mul_le_mul_of_nonneg_left (bound _ _ hb') (abs_nonneg (conjugate u)) nlinarith theorem scaled_quadratic_lattice_cover (L : ℝ) (hL : 1 ≤ L) (x y : ℝ) : ∃ z : CutRing, |ordinary z - x| ≤ 6 / L ∧ |conjugate z - y| ≤ 2 * L := by obtain ⟨k,hk,hk'⟩ := exists_nat_pow_near hL Real.one_lt_goldenRatio let u : CutRing := (cutTau - 1)^k have hτ : 0 < Real.goldenRatio^k := pow_pos Real.goldenRatio_pos _ have ho : ordinary u = (Real.goldenRatio^k)⁻¹ := by simp only [u,map_pow,map_sub,ordinary_cutTau,map_one] rw [← inv_pow] congr 1 rw [Real.inv_goldenRatio] linarith [Real.goldenRatio_add_goldenConj] have hc : |conjugate u| = Real.goldenRatio^k := by simp only [u,map_pow,map_sub,conjugate_cutTau,map_one,abs_pow] congr 1 rw [abs_of_neg (by linarith [Real.goldenConj_neg] : Real.goldenConj - 1 < 0)] linarith [Real.goldenRatio_add_goldenConj] have hu : u ≠ 0 := by intro h have he : ordinary u = 0 := by rw [h,map_zero] rw [ho] at he exact (inv_ne_zero hτ.ne') he obtain ⟨z,hz,hz'⟩ := quadratic_lattice_cover u hu x y refine ⟨z,hz.trans ?_,hz'.trans ?_⟩ · rw [ho,abs_of_pos (inv_pos.mpr hτ)] have hL0 : 0 < L := by linarith have hnear : L < 2*Real.goldenRatio^k := by rw [pow_succ] at hk' nlinarith [Real.goldenRatio_lt_two] apply (le_div_iff₀ hL0).mpr have he : (Real.goldenRatio^k)⁻¹ * Real.goldenRatio^k = 1 := inv_mul_cancel₀ hτ.ne' nlinarith [mul_lt_mul_of_pos_left hnear (inv_pos.mpr hτ)] · rw [hc] linarith theorem endpoint_separation {z : CutRing} (hz : z ≠ 0) {n : ℕ} (hn : 1 ≤ n) (ho : |ordinary z| ≤ 1) (hq : |(endpointLabel z : ℝ)| ≤ n) : 1 / (4 * (n : ℝ)) ≤ |ordinary z| := by have hsqrt : Real.sqrt 5 ≤ 3 := by nlinarith [Real.sq_sqrt (by norm_num : (0:ℝ) ≤ 5), Real.sqrt_nonneg 5] have hn' : (1:ℝ) ≤ n := by exact_mod_cast hn have he : conjugate z = ordinary z - (endpointLabel z : ℝ)*Real.sqrt 5 := by have h := (eq_div_iff (ne_of_gt (Real.sqrt_pos.mpr (by norm_num : (0:ℝ) < 5)))).mp (endpointLabel_embedding z) linarith have hc : |conjugate z| ≤ 4*(n:ℝ) := by rw [he] calc _ ≤ |ordinary z| + |(endpointLabel z:ℝ)*Real.sqrt 5| := by simpa using abs_sub_le (ordinary z) 0 ((endpointLabel z:ℝ)*Real.sqrt 5) _ = |ordinary z| + |(endpointLabel z:ℝ)| *Real.sqrt 5 := by rw [abs_mul,abs_of_nonneg (Real.sqrt_nonneg 5)] _ ≤ 1 + (n:ℝ)*3 := by gcongr _ ≤ _ := by linarith have hnorm := one_le_abs_cut_norm hz rw [abs_mul] at hnorm apply (div_le_iff₀ (by positivity : (0:ℝ) < 4*(n:ℝ))).mpr nlinarith [mul_le_mul_of_nonneg_left hc (abs_nonneg (ordinary z))] theorem coordinate_window_net (n : ℕ) (hn : 1 ≤ n) (q : ℤ) (t : ℝ) : ∃ z : CutRing, q ≤ endpointLabel z ∧ endpointLabel z < q + n ∧ |ordinary z - t| ≤ 100 / (n : ℝ) := by have hn0 : (0:ℝ) < n := by exact_mod_cast (lt_of_lt_of_le Nat.zero_lt_one hn) by_cases hn32 : 32 ≤ n · have hn32' : (32:ℝ) ≤ n := by exact_mod_cast hn32 let c : ℝ := (q:ℝ) + (n:ℝ)/2 obtain ⟨z,ho,hb⟩ := scaled_quadratic_lattice_cover ((n:ℝ)/16) (by linarith) t (t-c*Real.sqrt 5) have ho' : |ordinary z-t| ≤ 96/(n:ℝ) := by convert ho using 1; field_simp; ring have hb' : |conjugate z-(t-c*Real.sqrt 5)| ≤ (n:ℝ)/8 := by convert hb using 1; ring have hs : 1 ≤ Real.sqrt 5 := by nlinarith [Real.sq_sqrt (by norm_num : (0:ℝ) ≤ 5),Real.sqrt_nonneg 5] have hsmall : 96/(n:ℝ) ≤ (n:ℝ)/8 := by apply (div_le_iff₀ hn0).mpr nlinarith have hlabel : |(endpointLabel z:ℝ)-c| ≤ (n:ℝ)/4 := by have he : ((endpointLabel z:ℝ)-c)*Real.sqrt 5 = (ordinary z-t)-(conjugate z-(t-c*Real.sqrt 5)) := by have he := (eq_div_iff (ne_of_gt (lt_of_lt_of_le zero_lt_one hs))).mp (endpointLabel_embedding z) nlinarith [he] have htri : |(ordinary z-t)-(conjugate z-(t-c*Real.sqrt 5))| ≤ |ordinary z-t| + |conjugate z-(t-c*Real.sqrt 5)| := by simpa only [sub_zero,zero_sub,abs_neg] using abs_sub_le (ordinary z-t) 0 (conjugate z-(t-c*Real.sqrt 5)) rw [← he,abs_mul,abs_of_nonneg (le_trans zero_le_one hs)] at htri nlinarith [abs_nonneg ((endpointLabel z:ℝ)-c)] have hlow : (q:ℝ) ≤ (endpointLabel z:ℝ) := by have hh := (abs_le.mp hlabel).1 dsimp [c] at hh linarith have hhigh : (endpointLabel z:ℝ) < (q:ℝ)+(n:ℝ) := by have hh := (abs_le.mp hlabel).2 dsimp [c] at hh linarith refine ⟨z,by exact_mod_cast hlow,by exact_mod_cast hhigh,ho'.trans ?_⟩ apply div_le_div_of_nonneg_right (by norm_num) hn0.le · let z : CutRing := ⟨⌊t-(q:ℝ)*Real.goldenRatio⌋,q⟩ refine ⟨z,le_rfl,by change q < q+(n:ℤ); omega,?_⟩ have he : ordinary z-t = (⌊t-(q:ℝ)*Real.goldenRatio⌋:ℝ)- (t-(q:ℝ)*Real.goldenRatio) := by simp only [ordinary_apply,z]; ring rw [he] apply (floor_error _).trans apply (le_div_iff₀ hn0).mpr have : (n:ℝ) < 32 := by exact_mod_cast (lt_of_not_ge hn32) linarith end QuantitativeArithmetic end SimpleAmenable end end end OAI