import OAI.Geometry.LatticeCovering.BrokenCircuits namespace OAI section noncomputable section noncomputable section noncomputable section open MeasureTheory Filter Set open scoped Topology noncomputable section open MeasureTheory Filter Set open scoped Topology ENNReal noncomputable section noncomputable section noncomputable section noncomputable section noncomputable section noncomputable section noncomputable section noncomputable section noncomputable section noncomputable section noncomputable section noncomputable section noncomputable section noncomputable section namespace SingleLatticeCovering.RogersPreparation open Filter Topology lemma eventually_const_mul_rpow_lt {a : ℝ} (ha : a<1) (A : ℝ) : ∀ᶠ p : ℕ in atTop, A * (p : ℝ)^a < p := by have ht : Tendsto (fun p : ℕ => A * (p : ℝ)^(a-1)) atTop (𝓝 0) := by have h : Tendsto (fun p : ℕ => (p : ℝ)^(a-1)) atTop (𝓝 0) := by have hcast : Tendsto (fun p : ℕ => (p : ℝ)) atTop atTop := tendsto_natCast_atTop_atTop simpa only [Function.comp_def,neg_sub] using (tendsto_rpow_neg_atTop (sub_pos.mpr ha)).comp hcast simpa only [mul_zero] using h.const_mul A filter_upwards [ht.eventually (gt_mem_nhds (show (0:ℝ)<1 by norm_num)), eventually_ge_atTop 1] with p hp hp1 have hpR : (0:ℝ)

((integerGrid J hJ (primeScale d p)).card:ℝ)/(p:ℝ)) atTop (𝓝 (volume.real J)) := by simpa only [Function.comp_def,primeScale_pow hd] using (normalized_card_limit hJ hconv h0).comp (primeScale_tendsto hd) lemma eventually_prime_grid_bound {d k : ℕ} (hk : k < d) {J : Set (Fin d → ℝ)} (hJ : IsCompact J) : ∀ᶠ p : ℕ in atTop, ∃ B : ℕ, 1 ≤ B ∧ (∀ z ∈ integerGrid J hJ (primeScale d p), ∀ j, |z j| ≤ (B:ℤ)) ∧ k.factorial*B^k < p := by obtain ⟨C,hC,hbound⟩ := grid_uniform_bound hJ have h2C : 1 ≤ 2*C := by linarith filter_upwards [eventually_subcritical_minor_bound hk h2C, (primeScale_tendsto (by omega : 0 < d)).eventually (eventually_ge_atTop (1:ℝ))] with p hp ht let B : ℕ := ⌈C*primeScale d p⌉₊ have hCt : 1 ≤ C*primeScale d p := one_le_mul_of_one_le_of_one_le hC ht have hB : (1:ℝ) ≤ B := hCt.trans (Nat.le_ceil _) have hBC : (B:ℝ) ≤ 2*C*primeScale d p := by have hh := (Nat.ceil_lt_add_one (by linarith : 0 ≤ C*primeScale d p)).le dsimp only [B] nlinarith refine ⟨B,by exact_mod_cast hB,?_,?_⟩ · intro z hz j have hh := (hbound _ ht z hz j).trans (Nat.le_ceil _) exact_mod_cast hh · have hh : (k.factorial:ℝ)*(B:ℝ)^k < (p:ℝ) := (mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (Nat.cast_nonneg B) hBC k) (Nat.cast_nonneg _)).trans_lt hp.2 exact_mod_cast hh theorem eventually_prime_circuit_count_le {d r : ℕ} (hr : 0 < r) (hd : 2*(r+1) ≤ d) {J : Set (Fin d → ℝ)} (hJ : IsCompact J) (hconv : Convex ℝ J) (h0 : 0 ∈ J) {ε : ℝ} (hε : 0 < ε) : ∀ᶠ p : ℕ in atTop, ((circuits (integerGrid J hJ (primeScale d p)) r).card:ℝ)/(p:ℝ)^r < (SimplexYoung.affineCircuitVolumeSum (r:=r) J).toReal+ε := by obtain ⟨B,hB,hbound⟩ := grid_uniform_bound hJ have hh := eventually_circuit_count_le hr hd hJ hconv h0 (integerGrid J hJ) hB hbound (fun t ht z hz => (mem_integerGrid hJ (by linarith : t ≠ 0)).mp hz) hε simpa only [primeScale_pow_mul (by omega : 0 < d)] using (primeScale_tendsto (by omega : 0 < d)).eventually hh end SingleLatticeCovering.ConvexGrid noncomputable section namespace SingleLatticeCovering.GridProbability open Filter Topology open scoped BigOperators lemma cardinality_tendsto_atTop {M : ℕ → ℕ} {V : ℝ} (hV : 0 < V) (hM : Tendsto (fun p => (M p:ℝ)/(p:ℝ)) atTop (𝓝 V)) : Tendsto (fun p => (M p:ℝ)) atTop atTop := by have hh := hM.pos_mul_atTop hV (tendsto_natCast_atTop_atTop (R:=ℝ)) apply hh.congr' filter_upwards [eventually_ge_atTop 1] with p hp exact div_mul_cancel₀ _ (by exact_mod_cast (show p ≠ 0 by omega)) lemma normalized_choose_tendsto {M : ℕ → ℕ} {V : ℝ} (hV : 0 < V) (hM : Tendsto (fun p => (M p:ℝ)/(p:ℝ)) atTop (𝓝 V)) (j : ℕ) : Tendsto (fun p => ((M p).choose j:ℝ)/(p:ℝ)^j) atTop (𝓝 (V^j/(j.factorial:ℝ))) := by induction j with | zero => simp | succ j ih => have hinv : Tendsto (fun p : ℕ => (j:ℝ)/(p:ℝ)) atTop (𝓝 0) := by simpa only [mul_zero,div_eq_mul_inv,Pi.inv_apply] using (tendsto_natCast_atTop_atTop (R:=ℝ)).inv_tendsto_atTop.const_mul (j:ℝ) have ht := (ih.mul (hM.sub hinv)).div_const (j+1:ℝ) have hlim : (V^j/(j.factorial:ℝ))*(V-0)/(j+1:ℝ) = V^(j+1)/((j+1).factorial:ℝ) := by rw [Nat.factorial_succ,Nat.cast_mul,Nat.cast_add,Nat.cast_one,pow_succ,sub_zero] field_simp rw [hlim] at ht apply ht.congr' filter_upwards [(cardinality_tendsto_atTop hV hM).eventually (eventually_ge_atTop (j:ℝ)), eventually_ge_atTop 1] with p hpj hp1 have hjM : j ≤ M p := by exact_mod_cast hpj have hc : (((M p).choose (j+1):ℕ):ℝ)*(j+1:ℝ) = (((M p).choose j:ℕ):ℝ)*((M p:ℝ)-(j:ℝ)) := by exact_mod_cast Nat.choose_succ_right_eq (M p) j have hp0 : (p:ℝ) ≠ 0 := by exact_mod_cast (show p ≠ 0 by omega) have hj0 : (j+1:ℝ) ≠ 0 := by positivity field_simp [hp0,hj0,pow_succ] rw [←hc,pow_succ] ring lemma binomial_polynomial_limit {M : ℕ → ℕ} {V : ℝ} (hV : 0 < V) (hM : Tendsto (fun p => (M p:ℝ)/(p:ℝ)) atTop (𝓝 V)) (k : ℕ) : Tendsto (fun p => ∑ j ∈ Finset.range (k+1), ((M p).choose j:ℝ)*(-1:ℝ)^j*((p:ℝ)⁻¹)^j) atTop (𝓝 (∑ j ∈ Finset.range (k+1), (-1:ℝ)^j*V^j/(j.factorial:ℝ))) := by have hh := tendsto_finsetSum (Finset.range (k+1)) (fun j hj => (normalized_choose_tendsto hV hM j).const_mul ((-1:ℝ)^j)) simpa only [inv_pow,div_eq_mul_inv,mul_assoc,mul_left_comm,mul_comm] using hh lemma binomial_envelope_le_exp (M p : ℕ) : (1+(p:ℝ)⁻¹)^M ≤ Real.exp ((M:ℝ)/(p:ℝ)) := by calc _ ≤ (Real.exp ((p:ℝ)⁻¹))^M := pow_le_pow_left₀ (by positivity) (by linarith [Real.add_one_le_exp ((p:ℝ)⁻¹)]) M _ = _ := by rw [←Real.exp_nat_mul]; congr 1 lemma upper_envelope_limit {α : Type*} {l : Filter α} {f : α → ℝ} {C : ℝ} (hf : ∀ ε : ℝ, 0 < ε → ∀ᶠ x in l, f x < C+ε) : Tendsto (fun x => max (f x) C) l (𝓝 C) := by apply tendsto_order.mpr constructor · intro b hb exact Filter.Eventually.of_forall (fun x => hb.trans_le (le_max_right _ _)) · intro b hb filter_upwards [hf (b-C) (sub_pos.mpr hb)] with x hx exact max_lt (by linarith) hb end SingleLatticeCovering.GridProbability namespace SingleLatticeCovering.ConvexGrid open MeasureTheory Filter Topology AffineCircuit GridProbability BrokenCircuit PrimeCircuit open scoped BigOperators def primeGridVoid {d : ℕ} (J : Set (Fin d → ℝ)) (hJ : IsCompact J) (p : ℕ) [Fact p.Prime] : ℝ := vectorAverage (fun a : Fin (d+1) → ZMod p => avoidanceIndicator (fun z : integerGrid J hJ (primeScale d p) => homogeneousRow (p:=p) z.val) (dotProductBilin (ZMod p) (ZMod p) a)) def continuumCircuitError {d : ℕ} (J : Set (Fin d → ℝ)) (k : ℕ) : ℝ := Real.exp (volume.real J) * ∑ r ∈ Finset.Icc 2 k, (SimplexYoung.affineCircuitVolumeSum (r:=r) J).toReal/(r.factorial:ℝ) theorem eventually_primeGridVoid_le {d q : ℕ} (hq : 0 < q) (hd : 2*(2*q+1) ≤ d) {J : Set (Fin d → ℝ)} (hJ : IsCompact J) (hconv : Convex ℝ J) (h0 : 0 ∈ J) (hV : 0 < volume.real J) {ε : ℝ} (hε : 0 < ε) : ∃ N : ℕ, ∀ (p : ℕ) [Fact p.Prime], N ≤ p → primeGridVoid J hJ p ≤ (∑ j ∈ Finset.range (2*q+1), (-1:ℝ)^j*(volume.real J)^j/(j.factorial:ℝ)) + continuumCircuitError J (2*q)+ε := by classical let S (p : ℕ) := integerGrid J hJ (primeScale d p) let C (r : ℕ) := (SimplexYoung.affineCircuitVolumeSum (r:=r) J).toReal let U (p r : ℕ) := max (((circuits (S p) r).card:ℝ)/(p:ℝ)^r) (C r) have hU (r : ℕ) (hr : r ∈ Finset.Icc 2 (2*q)) : Tendsto (fun p => U p r) atTop (𝓝 (C r)) := by apply upper_envelope_limit intro δ hδ exact eventually_prime_circuit_count_le (by have := (Finset.mem_Icc.mp hr).1; omega) (by have := (Finset.mem_Icc.mp hr).2; omega) hJ hconv h0 hδ have hsum := tendsto_finsetSum (Finset.Icc 2 (2*q)) (fun r hr => (hU r hr).div_const (r.factorial:ℝ)) have hmass := prime_grid_density (by omega : 0 < d) hJ hconv h0 have hlim := (binomial_polynomial_limit hV hmass (2*q)).add (((Real.continuous_exp.tendsto _).comp hmass).mul hsum) have hless : (∑ j ∈ Finset.range (2*q+1), (-1:ℝ)^j*(volume.real J)^j/(j.factorial:ℝ)) + Real.exp (volume.real J) * ∑ r ∈ Finset.Icc 2 (2*q), C r/(r.factorial:ℝ) < (∑ j ∈ Finset.range (2*q+1), (-1:ℝ)^j*(volume.real J)^j/(j.factorial:ℝ)) + continuumCircuitError J (2*q)+ε := by change _ + continuumCircuitError J (2*q) < _ exact lt_add_of_pos_right _ hε have hevent := hlim.eventually (gt_mem_nhds hless) have hminor := eventually_prime_grid_bound (k:=2*q+1) (by omega) hJ obtain ⟨N,hN⟩ := eventually_atTop.mp (hevent.and hminor) refine ⟨N,?_⟩ intro p hpPrime hpN obtain ⟨he,⟨B,hB,hSB,hpB⟩⟩ := hN p hpN let : LinearOrder (S p) := (Fintype.equivFin (S p)).linearOrder have hp2 : 2*B < p := by have hf : 2 ≤ (2*q+1).factorial := by simpa using (Nat.factorial_le (show 2 ≤ 2*q+1 by omega)) have hb : B ≤ B^(2*q+1) := by simpa only [pow_one] using Nat.pow_le_pow_right hB (show 1 ≤ 2*q+1 by omega) exact (Nat.mul_le_mul hf hb).trans_lt hpB let v : S p → Fin (d+1) → ZMod p := fun z => homogeneousRow z.val have hfin := finite_field_void_upper v q simp only [ZMod.card,Fintype.card_coe] at hfin have herr := finiteCircuitError_le (k:=2*q) (S p) hB hSB hp2 hpB have hsn : 0 ≤ ∑ r ∈ Finset.Icc 2 (2*q), ((circuits (S p) r).card:ℝ)/(p:ℝ)^r/(r.factorial:ℝ) := by apply Finset.sum_nonneg intro r hr exact div_nonneg (div_nonneg (Nat.cast_nonneg _) (pow_nonneg (Nat.cast_nonneg _) _)) (Nat.cast_nonneg _) have hreplace : ∑ r ∈ Finset.Icc 2 (2*q), ((circuits (S p) r).card:ℝ)/(p:ℝ)^r/(r.factorial:ℝ) ≤ ∑ r ∈ Finset.Icc 2 (2*q), U p r/(r.factorial:ℝ) := by apply Finset.sum_le_sum intro r hr exact div_le_div_of_nonneg_right (le_max_left _ _) (Nat.cast_nonneg _) have herr' := herr.trans ((mul_le_mul_of_nonneg_right (binomial_envelope_le_exp (S p).card p) hsn).trans (mul_le_mul_of_nonneg_left hreplace (Real.exp_pos _).le)) exact (hfin.trans (add_le_add le_rfl herr')).trans he.le end SingleLatticeCovering.ConvexGrid namespace SingleLatticeCovering.SimplexYoung open scoped BigOperators lemma simplexConstant_sq_ratio {r : ℕ} (hr : 0 < r) : (simplexConstant r)^2 = ((r+1:ℕ):ℝ)^r / ((r:ℝ)^r * (r+1:ℕ)) := by have h : 0 < (r:ℝ) := Nat.cast_pos.mpr hr unfold simplexConstant rw [div_pow, ←pow_mul, mul_comm r 2, pow_mul, inv_pow, Real.sq_sqrt (by positivity), Real.sq_sqrt (by positivity), inv_div, div_pow] ring lemma simplexConstant_succ_sq {r : ℕ} (hr : 0 < r) : (simplexConstant (r+1))^2 = (simplexConstant r)^2 * (1 - 1 / (((r+1:ℕ):ℝ)^2))^r := by have h : 0 < (r:ℝ) := Nat.cast_pos.mpr hr have h1 : ((r+1:ℕ):ℝ) ≠ 0 := by positivity have h2 : ((r+1+1:ℕ):ℝ) ≠ 0 := by positivity rw [simplexConstant_sq_ratio (by omega), simplexConstant_sq_ratio hr] have hc : (1 - 1 / (((r+1:ℕ):ℝ)^2)) = (r:ℝ) * (r+2) / (((r+1:ℕ):ℝ)^2) := by push_cast field_simp ring rw [hc, div_pow, mul_pow, pow_succ, pow_succ, ←pow_mul] push_cast rw [show 2*r=r+r by omega, pow_add] field_simp ring lemma simplexConstant_succ_le {r : ℕ} (hr : 0 < r) : simplexConstant (r+1) ≤ simplexConstant r * Real.exp (-(r:ℝ)/(2*((r+1:ℕ):ℝ)^2)) := by have hbase : 0 ≤ 1 - 1 / (((r+1:ℕ):ℝ)^2) := by have h : (1:ℝ) ≤ (r+1:ℕ) := by exact_mod_cast (show 1 ≤ r+1 by omega) have hsq : (1:ℝ) ≤ ((r+1:ℕ):ℝ)^2 := by nlinarith have hi := (div_le_one (by positivity : (0:ℝ)<((r+1:ℕ):ℝ)^2)).mpr hsq linarith have hexp : (1 - 1 / (((r+1:ℕ):ℝ)^2))^r ≤ Real.exp (-(r:ℝ)/((r+1:ℕ):ℝ)^2) := by calc _ ≤ (Real.exp (-(1 / (((r+1:ℕ):ℝ)^2))))^r := pow_le_pow_left₀ hbase (by linarith [Real.add_one_le_exp (-(1/(((r+1:ℕ):ℝ)^2)))]) r _ = _ := by rw [←Real.exp_nat_mul]; congr 1; ring apply (sq_le_sq₀ (simplexConstant_nonneg _) (mul_nonneg (simplexConstant_nonneg _) (Real.exp_pos _).le)).mp rw [simplexConstant_succ_sq hr, mul_pow, ←Real.exp_nat_mul] have he : (2:ℝ) * (-(r:ℝ)/(2*((r+1:ℕ):ℝ)^2)) = -(r:ℝ)/((r+1:ℕ):ℝ)^2 := by ring norm_num only [Nat.cast_ofNat] rw [he] exact mul_le_mul_of_nonneg_left hexp (sq_nonneg _) def circuitTerm (d : ℕ) (V : ℝ) (r : ℕ) : ℝ := (4:ℝ)^r * (simplexConstant r)^d * V^r / (r.factorial:ℝ) lemma circuitTerm_nonneg (d r : ℕ) {V : ℝ} (hV : 0 ≤ V) : 0 ≤ circuitTerm d V r := by unfold circuitTerm exact div_nonneg (mul_nonneg (mul_nonneg (by positivity) (pow_nonneg (simplexConstant_nonneg _) _)) (pow_nonneg hV _)) (by positivity) lemma circuitTerm_succ_le {d r : ℕ} (hr : 3 ≤ r) {V : ℝ} (hV : 0 ≤ V) (hVd : V ≤ (d:ℝ)/6) : circuitTerm d V (r+1) ≤ (3/4:ℝ) * circuitTerm d V r := by have hr0 : 0 < r := by omega have hden : (0:ℝ) < (r+1:ℕ) := by positivity have hgeo := pow_le_pow_left₀ (simplexConstant_nonneg _) (simplexConstant_succ_le hr0) d rw [mul_pow, ←Real.exp_nat_mul] at hgeo have hexp : -(d:ℝ) * (r:ℝ)/(2*((r+1:ℕ):ℝ)^2) ≤ -(3:ℝ)/8 * ((d:ℝ)/(r+1:ℕ)) := by have hrR : (3:ℝ) ≤ r := by exact_mod_cast hr apply (div_le_iff₀ (by positivity : (0:ℝ)<2*((r+1:ℕ):ℝ)^2)).mpr have he : (-(3:ℝ)/8 * ((d:ℝ)/(r+1:ℕ))) * (2*((r+1:ℕ):ℝ)^2) = -(3:ℝ)/4 * (d:ℝ) * ((r+1:ℕ):ℝ) := by field_simp; ring rw [he] push_cast nlinarith [mul_nonneg (Nat.cast_nonneg d : (0:ℝ)≤d) (sub_nonneg.mpr hrR)] have heq : (d:ℝ) * (-(r:ℝ)/(2*((r+1:ℕ):ℝ)^2)) = -(d:ℝ) * (r:ℝ)/(2*((r+1:ℕ):ℝ)^2) := by ring rw [heq] at hgeo have hgeo' := hgeo.trans (mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr hexp) (pow_nonneg (simplexConstant_nonneg _) _)) have hb : (4*V/(r+1:ℕ))*Real.exp (-(3:ℝ)/8*((d:ℝ)/(r+1:ℕ))) ≤ 3/4 := by have ht := Real.mul_exp_neg_le_exp_neg_one ((3:ℝ)/8*((d:ℝ)/(r+1:ℕ))) have he : Real.exp (-1) ≤ (27/64:ℝ) := by rw [Real.exp_neg, inv_eq_one_div] apply (div_le_iff₀ (Real.exp_pos _)).mpr nlinarith [Real.exp_one_gt_d9] calc _ ≤ ((2:ℝ)/3*((d:ℝ)/(r+1:ℕ)))*Real.exp (-(3:ℝ)/8*((d:ℝ)/(r+1:ℕ))) := by apply mul_le_mul_of_nonneg_right _ (Real.exp_pos _).le apply (div_le_iff₀ hden).mpr have heq : ((2:ℝ)/3*((d:ℝ)/(r+1:ℕ)))*((r+1:ℕ):ℝ) = (2:ℝ)/3*d := by field_simp rw [heq] linarith _ = (16/9:ℝ)*(((3:ℝ)/8*((d:ℝ)/(r+1:ℕ)))* Real.exp (-((3:ℝ)/8*((d:ℝ)/(r+1:ℕ))))) := by rw [show -(3:ℝ)/8*((d:ℝ)/(r+1:ℕ)) = -((3:ℝ)/8*((d:ℝ)/(r+1:ℕ))) by ring] ring _ ≤ (16/9:ℝ)*Real.exp (-1) := mul_le_mul_of_nonneg_left ht (by norm_num) _ ≤ 3/4 := by linarith unfold circuitTerm rw [pow_succ, pow_succ, Nat.factorial_succ, Nat.cast_mul, Nat.cast_add, Nat.cast_one] calc _ ≤ ((4:ℝ)^r*4) * ((simplexConstant r)^d * Real.exp (-(3:ℝ)/8*((d:ℝ)/(r+1:ℕ)))) * (V^r*V) / (((r:ℝ)+1)*(r.factorial:ℝ)) := by gcongr _ = ((4*V/(r+1:ℕ))*Real.exp (-(3:ℝ)/8*((d:ℝ)/(r+1:ℕ)))) * ((4:ℝ)^r*(simplexConstant r)^d*V^r/(r.factorial:ℝ)) := by push_cast; field_simp _ ≤ _ := mul_le_mul_of_nonneg_right hb (circuitTerm_nonneg d r hV) lemma circuitTerm_add_three_le {d : ℕ} {V : ℝ} (hV : 0 ≤ V) (hVd : V ≤ (d:ℝ)/6) (s : ℕ) : circuitTerm d V (s+3) ≤ (3/4:ℝ)^s * circuitTerm d V 3 := by induction s with | zero => simp | succ s ih => calc _ ≤ (3/4:ℝ) * circuitTerm d V (s+3) := circuitTerm_succ_le (by omega) hV hVd _ ≤ (3/4:ℝ) * ((3/4:ℝ)^s * circuitTerm d V 3) := mul_le_mul_of_nonneg_left ih (by norm_num) _ = _ := by rw [pow_succ]; ring theorem tsum_circuitTerm_le {d : ℕ} {V : ℝ} (hV : 0 ≤ V) (hVd : V ≤ (d:ℝ)/6) : ∑' s : ℕ, ENNReal.ofReal (circuitTerm d V (s+3)) ≤ ENNReal.ofReal (4 * circuitTerm d V 3) := by calc _ ≤ ∑' s : ℕ, ENNReal.ofReal ((3/4:ℝ)^s * circuitTerm d V 3) := ENNReal.tsum_le_tsum (fun s => ENNReal.ofReal_le_ofReal (circuitTerm_add_three_le hV hVd s)) _ = ENNReal.ofReal (4 * circuitTerm d V 3) := by rw [←ENNReal.ofReal_tsum_of_nonneg (fun s => mul_nonneg (by positivity) (circuitTerm_nonneg _ _ hV)) ((summable_geometric_of_norm_lt_one (by norm_num : ‖(3/4:ℝ)‖ < 1)).mul_right _)] rw [tsum_mul_right, tsum_geometric_of_norm_lt_one (by norm_num : ‖(3/4:ℝ)‖ < 1)] norm_num end SingleLatticeCovering.SimplexYoung namespace SingleLatticeCovering.SimplexYoung lemma sharpParallelogramConstant_pos : 0 < (4 * (Real.sqrt 3)⁻¹^3 : ℝ) := by positivity lemma sharpParallelogramConstant_sq : (4 * (Real.sqrt 3)⁻¹^3 : ℝ)^2 = 16/27 := by have hs : (Real.sqrt 3)^2 = (3:ℝ) := Real.sq_sqrt (by norm_num) calc _ = 16 / (Real.sqrt 3)^6 := by field_simp; ring _ = _ := by rw [show 6 = 2*3 from rfl, pow_mul, hs]; norm_num lemma sharpParallelogramConstant_exp (d : ℕ) : (4 * (Real.sqrt 3)⁻¹^3 : ℝ)^d = Real.exp (-(d:ℝ)/2 * Real.log (27/16:ℝ)) := by have hp := sharpParallelogramConstant_pos have hlog := congrArg Real.log sharpParallelogramConstant_sq rw [Real.log_pow] at hlog have hi : Real.log (16/27:ℝ) = -Real.log (27/16:ℝ) := by rw [← Real.log_inv]; norm_num rw [hi] at hlog rw [← Real.exp_log hp, ← Real.exp_nat_mul] congr 1 push_cast at hlog nlinarith def sourceEta (d : ℕ) : ℝ := (d:ℝ)/4 * Real.log (27/16:ℝ) - 3*Real.log (d:ℝ) lemma sourceLog_pos : 0 < Real.log (27/16:ℝ) := Real.log_pos (by norm_num) lemma sourceLog_le_one : Real.log (27/16:ℝ) ≤ 1 := by have h := Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<27/16) linarith lemma sourceEta_le_dim {d : ℕ} (hd : 1 ≤ d) : sourceEta d ≤ d := by have hdR : (1:ℝ) ≤ d := by exact_mod_cast hd have hl : 0 ≤ Real.log (d:ℝ) := Real.log_nonneg hdR have hx := sourceLog_le_one dsimp [sourceEta] nlinarith lemma log_27_16_le : Real.log (27/16 : ℝ)≤8/15 := by apply (Real.log_le_iff_le_exp (by norm_num : (0 : ℝ)<27/16)).mpr have h := Real.sum_le_exp_of_nonneg (by norm_num : (0 : ℝ)≤8/15) 4 norm_num [Finset.sum_range_succ,Nat.factorial] at h linarith lemma sourceEta_le_linear (d : ℕ) : sourceEta d≤(2/15 : ℝ)*d := by have h := mul_le_mul_of_nonneg_left log_27_16_le (show (0 : ℝ)≤(d : ℝ)/4 by positivity) have hlog := Real.log_natCast_nonneg d unfold sourceEta nlinarith lemma simplexConstant_three : simplexConstant 3 = 4 * (Real.sqrt 3)⁻¹^3 := by unfold simplexConstant norm_num [Real.sqrt_div, div_pow] ring lemma circuitTerm_three_source {d : ℕ} (hd : 1 ≤ d) {V : ℝ} (hV : 0 ≤ V) (hVeta : V ≤ sourceEta d) : Real.exp V * (4 * circuitTerm d V 3) ≤ (128/3:ℝ) * Real.exp (-sourceEta d) := by have hdR : (0:ℝ) < d := by exact_mod_cast hd have hVd := hVeta.trans (sourceEta_le_dim hd) have hpow : V^3 ≤ (d:ℝ)^3 := pow_le_pow_left₀ hV hVd _ have h : Real.exp V * ((4*(Real.sqrt 3)⁻¹^3:ℝ)^d * V^3) ≤ Real.exp (-sourceEta d) := by rw [sharpParallelogramConstant_exp] calc _ ≤ Real.exp V * (Real.exp (-(d:ℝ)/2*Real.log (27/16:ℝ)) * (d:ℝ)^3) := by gcongr _ = Real.exp (V-(d:ℝ)/2*Real.log (27/16:ℝ)+3*Real.log (d:ℝ)) := by have he : (d:ℝ)^3 = Real.exp (3*Real.log (d:ℝ)) := by simpa only [Nat.cast_ofNat, Real.exp_log hdR] using (Real.exp_nat_mul (Real.log (d:ℝ)) 3).symm rw [he, ←Real.exp_add, ←Real.exp_add] congr 1 ring _ ≤ _ := by apply Real.exp_le_exp.mpr have hl := Real.log_nonneg (show (1:ℝ)≤d by exact_mod_cast hd) dsimp [sourceEta] at hVeta ⊢ linarith unfold circuitTerm rw [simplexConstant_three] norm_num only [show (4:ℝ)^3=64 by norm_num, show ((3:ℕ).factorial:ℝ)=6 by norm_num] have hh := mul_le_mul_of_nonneg_left h (by norm_num : (0:ℝ)≤128/3) calc _ = (128/3:ℝ) * (Real.exp V * ((4*(Real.sqrt 3)⁻¹^3:ℝ)^d * V^3)) := by ring _ ≤ _ := hh theorem circuit_series_at_sourceEta {d : ℕ} (hd : 1 ≤ d) {V : ℝ} (hV : 0 ≤ V) (hVeta : V ≤ sourceEta d) : ENNReal.ofReal (Real.exp V) * (∑' s : ℕ, ENNReal.ofReal (circuitTerm d V (s+3))) ≤ ENNReal.ofReal ((128/3:ℝ)*Real.exp (-sourceEta d)) := by have hVd : V ≤ (d:ℝ)/6 := by have h := hVeta.trans (sourceEta_le_linear d) nlinarith [(Nat.cast_nonneg d : (0:ℝ)≤d)] calc _ ≤ ENNReal.ofReal (Real.exp V) * ENNReal.ofReal (4*circuitTerm d V 3) := mul_le_mul_right (tsum_circuitTerm_le hV hVd) _ _ = ENNReal.ofReal (Real.exp V * (4*circuitTerm d V 3)) := (ENNReal.ofReal_mul (Real.exp_pos _).le).symm _ ≤ _ := ENNReal.ofReal_le_ofReal (circuitTerm_three_source hd hV hVeta) end SingleLatticeCovering.SimplexYoung noncomputable section namespace SingleLatticeCovering.RogersPreparation open Finset Set open scoped BigOperators def poissonPartial (u : ℝ) (k : ℕ) : ℝ := ∑ j ∈ range (k+1), (-u)^j/(j.factorial : ℝ) lemma poissonPartial_zero (k : ℕ) : poissonPartial 0 k=1 := by unfold poissonPartial rw [sum_range_succ'] simp lemma taylor_exp_neg_eq {u : ℝ} (hu : 0 < u) (k : ℕ) : taylorWithinEval Real.exp k (uIcc 0 (-u)) 0 (-u)=poissonPartial u k := by have hne : (0 : ℝ)≠-u := by linarith have hd (j : ℕ) : iteratedDerivWithin j Real.exp (uIcc 0 (-u)) 0=1 := by rw [iteratedDerivWithin_eq_iteratedDeriv (uniqueDiffOn_uIcc hne) Real.contDiff_exp.contDiffAt (left_mem_uIcc),iteratedDeriv_eq_iterate, Real.iter_deriv_exp,Real.exp_zero] rw [taylor_within_apply] unfold poissonPartial apply Finset.sum_congr rfl intro j hj rw [hd] simp [smul_eq_mul,div_eq_mul_inv,mul_comm] theorem poissonPartial_error {u : ℝ} (hu : 0 ≤ u) (k : ℕ) : |poissonPartial u k-Real.exp (-u)| ≤ u^(k+1)/(k+1).factorial := by rcases hu.eq_or_lt with he | hu · subst u simp [poissonPartial_zero] have hne : (0 : ℝ)≠-u := by linarith obtain ⟨c,hc,he⟩ := taylor_mean_remainder_lagrange_iteratedDeriv (n := k) hne Real.contDiff_exp.contDiffOn rw [taylor_exp_neg_eq hu k,iteratedDeriv_eq_iterate,Real.iter_deriv_exp,sub_zero] at he have hc0 : c ≤ 0 := by have h := hc.2 simpa only [max_eq_left (neg_nonpos.mpr hu.le)] using h.le have hec : Real.exp c ≤ 1 := by simpa using Real.exp_le_exp.mpr hc0 rw [abs_sub_comm,he,abs_div,abs_mul,abs_pow,abs_neg,abs_of_nonneg hu.le, abs_of_pos (Real.exp_pos c),abs_of_nonneg (by positivity : (0 : ℝ)≤(k+1).factorial)] apply div_le_div_of_nonneg_right _ (by positivity) exact mul_le_of_le_one_left (by positivity) hec theorem poissonPartial_upper {u : ℝ} (hu : 0 ≤ u) (k : ℕ) : poissonPartial u k ≤ Real.exp (-u)+u^(k+1)/(k+1).factorial := by have := le_trans (le_abs_self (poissonPartial u k-Real.exp (-u))) (poissonPartial_error hu k) linarith end SingleLatticeCovering.RogersPreparation namespace SingleLatticeCovering.SimplexYoung open scoped BigOperators lemma log_four_thirds_ge : (2/7 : ℝ)≤Real.log (4/3 : ℝ) := by apply (Real.le_log_iff_exp_le (by norm_num : (0 : ℝ)<4/3)).mpr have h := Real.exp_le_two_add_div_two_sub (by norm_num : (0 : ℝ)≤2/7) (by norm_num : (2/7 : ℝ)<2) norm_num at h ⊢ exact h lemma log_three_quarters_le : Real.log (3/4 : ℝ)≤-(2/7 : ℝ) := by have he : (3/4 : ℝ)=(4/3 : ℝ)⁻¹ := by norm_num rw [he,Real.log_inv] linarith [log_four_thirds_ge] lemma factorial_quotient_chernoff {u : ℝ} (hu : 0≤u) {q : ℕ} (hq : 0