| 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 |
|
|
