File size: 5,514 Bytes
d4317d7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 | import OAI.MeasureTheory.Falconer.Geometry.FiniteProjectionSublevels
import OAI.MeasureTheory.Falconer.Recursion.FiniteCriticalPotential
import OAI.MeasureTheory.Falconer.Recursion.FiniteCountingPotential
namespace OAI
/-! The finite projected-energy average used for the good projections in
Orponen--Shmerkin, Appendix A.4. The critical directional loss is linear in
the dyadic depth; the planar point potential is strictly subcritical. -/
namespace Falconer
open MeasureTheory
open scoped BigOperators
attribute [local instance] Classical.propDecidable
theorem finite_projection_potential (N : ℕ) {s K : ℝ}
(hs : 0 ≤ s) (hK : 1 ≤ K) (I : Finset ℝ) (hI : ∀ c ∈ I, |c| ≤ 1)
(hball : ∀ u r : ℝ, 1/(2:ℝ)^N ≤ r → r ≤ 1 →
scalarBallMass I u r ≤ K*r^s*(I.card:ℝ))
(p q : Space 2) (hdiam : dist p q ≤ 4) :
(∑ c ∈ I, (1/max |lineParameterProjection c p-lineParameterProjection c q| (1/(2:ℝ)^N))^s) ≤
(I.card:ℝ)+(2:ℝ)^s*(1+(N:ℝ)*(2:ℝ)^s)*
(K*(8/max (dist p q) (1/(2:ℝ)^N))^s*(I.card:ℝ)) := by
let δ := 1/(2:ℝ)^N
let D := max (dist p q) δ
let C := K*(8/D)^s*(I.card:ℝ)
let a := fun c => |lineParameterProjection c p-lineParameterProjection c q|
have hδ : 0 < δ := by dsimp [δ]; positivity
have hD : 0 < D := hδ.trans_le (le_max_right _ _)
have hC : 0 ≤ C := by dsimp [C]; positivity
have hcounts (j : ℕ) (hj : j ≤ N) :
((I.filter fun c => a c < 1/(2:ℝ)^j).card:ℝ) ≤ C*(1/(2:ℝ)^j)^s := by
have hr : δ ≤ 1/(2:ℝ)^j := by
apply one_div_le_one_div_of_le (by positivity)
exact pow_le_pow_right₀ (by norm_num) hj
have hh := finite_projection_sublevel hδ hr hs hK I hI hball p q hdiam
have hsub : I.filter (fun c => a c < 1/(2:ℝ)^j) ⊆
I.filter (fun c => a c ≤ 1/(2:ℝ)^j) :=
by
intro c hc
exact Finset.mem_filter.mpr ⟨(Finset.mem_filter.mp hc).1,(Finset.mem_filter.mp hc).2.le⟩
apply (Nat.cast_le.mpr (Finset.card_le_card hsub)).trans
apply hh.trans_eq
change K*(8*(1/(2:ℝ)^j)/D)^s*(I.card:ℝ) = _
rw [show 8*(1/(2:ℝ)^j)/D = (8/D)*(1/(2:ℝ)^j) by ring,
Real.mul_rpow (by positivity) (by positivity)]
dsimp [C]
ring
have ha : Measurable a := by dsimp [a,lineParameterProjection]; fun_prop
have hh := finite_critical_statistic I a ha (fun _ => abs_nonneg _) N hs hC hcounts
convert hh using 1
dsimp [C,D,δ]
ring
noncomputable def planarProjectionEnergy (N : ℕ) (s c : ℝ) (P : Finset (Space 2)) : ℝ :=
∑ p ∈ P, ∑ q ∈ P,
(1/max |lineParameterProjection c p-lineParameterProjection c q| (1/(2:ℝ)^N))^s
theorem finite_projection_energy_average (N : ℕ) {s t KI KP : ℝ}
(hs : 0 ≤ s) (hst : s < t) (hKI : 1 ≤ KI) (hKP : 1 ≤ KP)
(I : Finset ℝ) (P : Finset (Space 2))
(hI : ∀ c ∈ I, |c| ≤ 1) (hP : InBoundedLineChart P)
(hIball : ∀ u r : ℝ, 1/(2:ℝ)^N ≤ r → r ≤ 1 →
scalarBallMass I u r ≤ KI*r^s*(I.card:ℝ))
(hPball : NormalizedFiniteBallBound (1/(2:ℝ)^N) t KP P) :
(∑ c ∈ I, planarProjectionEnergy N s c P) ≤
(I.card:ℝ)*(P.card:ℝ)^2*
(1+((2:ℝ)^s*(1+(N:ℝ)*(2:ℝ)^s)*KI*(8:ℝ)^s)*KP*planarPotentialConstant s t) := by
let A := (2:ℝ)^s*(1+(N:ℝ)*(2:ℝ)^s)*KI*(8:ℝ)^s
let k := fun p q : Space 2 => (1/max (dist p q) (1/(2:ℝ)^N))^s
have hA : 0 ≤ A := by dsimp [A]; positivity
have hpair (p : Space 2) (hp : p ∈ P) (q : Space 2) (hq : q ∈ P) :
(∑ c ∈ I, (1/max |lineParameterProjection c p-lineParameterProjection c q| (1/(2:ℝ)^N))^s) ≤
(I.card:ℝ)+A*(I.card:ℝ)*k p q := by
have hd : dist p q ≤ 4 := by
have h0 := abs_sub (p 0) (q 0)
have h1 := abs_sub (p 1) (q 1)
have hg := planar_dist_le_coordinate_sum p q
linarith only [h0,h1,hg,(hP p hp).1,(hP p hp).2,(hP q hq).1,(hP q hq).2]
have hh := finite_projection_potential N hs hKI I hI hIball p q hd
apply hh.trans_eq
rw [show 8/max (dist p q) (1/(2:ℝ)^N) = 8*(1/max (dist p q) (1/(2:ℝ)^N)) by ring,
Real.mul_rpow (by norm_num : (0:ℝ) ≤ 8) (by positivity)]
dsimp [A,k]
ring
have hpoint (p : Space 2) : (∑ q ∈ P, k p q) ≤ KP*planarPotentialConstant s t*(P.card:ℝ) :=
finite_dyadic_truncated_potential_real P p N hKP hs hst hPball
have hpotential : (∑ p ∈ P, ∑ q ∈ P, k p q) ≤ KP*planarPotentialConstant s t*(P.card:ℝ)^2 := by
calc
_ ≤ ∑ _p ∈ P, KP*planarPotentialConstant s t*(P.card:ℝ) := Finset.sum_le_sum fun p _ => hpoint p
_ = _ := by simp; ring
calc
_ = ∑ p ∈ P, ∑ q ∈ P, ∑ c ∈ I,
(1/max |lineParameterProjection c p-lineParameterProjection c q| (1/(2:ℝ)^N))^s := by
unfold planarProjectionEnergy
rw [Finset.sum_comm]
apply Finset.sum_congr rfl
intro p _
rw [Finset.sum_comm]
_ ≤ ∑ p ∈ P, ∑ q ∈ P, ((I.card:ℝ)+A*(I.card:ℝ)*k p q) :=
Finset.sum_le_sum fun p hp => Finset.sum_le_sum fun q hq => hpair p hp q hq
_ = (I.card:ℝ)*(P.card:ℝ)^2+A*(I.card:ℝ)*(∑ p ∈ P, ∑ q ∈ P, k p q) := by
simp only [Finset.sum_add_distrib,Finset.sum_const,nsmul_eq_mul,← Finset.mul_sum]
ring
_ ≤ (I.card:ℝ)*(P.card:ℝ)^2+A*(I.card:ℝ)*(KP*planarPotentialConstant s t*(P.card:ℝ)^2) :=
add_le_add_right (mul_le_mul_of_nonneg_left hpotential (mul_nonneg hA (Nat.cast_nonneg I.card))) _
_ = _ := by dsimp [A]; ring
end Falconer
end OAI
|