File size: 3,947 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 | import OAI.MeasureTheory.Falconer.Recursion.CriticalTruncatedPotential
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
namespace OAI
/-! The critical dyadic potential of a finite scalar statistic, with one
explicit contribution per scale. This is used for the average projected
energy of a planar point set. -/
namespace Falconer
open Set MeasureTheory
open scoped ENNReal BigOperators
attribute [local instance] Classical.propDecidable
noncomputable def scalarCountingMeasure (I : Finset ℝ) : Measure ℝ :=
∑ c ∈ I, Measure.dirac c
theorem scalarCountingMeasure_apply (I : Finset ℝ) (A : Set ℝ) :
scalarCountingMeasure I A = (I.filter (fun c => c ∈ A)).card := by
rw [scalarCountingMeasure,Measure.finsetSum_apply]
simp_rw [Measure.dirac_apply,Set.indicator_apply,Pi.one_apply]
rw [← Finset.sum_filter]
simp
theorem scalarCountingMeasure_lintegral (I : Finset ℝ) (f : ℝ → ℝ≥0∞) :
(∫⁻ c, f c ∂scalarCountingMeasure I) = ∑ c ∈ I, f c := by
rw [scalarCountingMeasure,lintegral_finsetSum_measure]
simp only [lintegral_dirac]
theorem finite_critical_statistic (I : Finset ℝ) (a : ℝ → ℝ)
(ha : Measurable a) (ha0 : ∀ c, 0 ≤ a c) (N : ℕ) {s C : ℝ}
(hs : 0 ≤ s) (hC : 0 ≤ C)
(hsmall : ∀ j : ℕ, j ≤ N →
((I.filter fun c => a c < 1/(2:ℝ)^j).card:ℝ) ≤ C*(1/(2:ℝ)^j)^s) :
(∑ c ∈ I, (1/max (a c) (1/(2:ℝ)^N))^s) ≤
(I.card:ℝ)+(2:ℝ)^s*(C+(N:ℝ)*(C*(2:ℝ)^s)) := by
let μ := scalarCountingMeasure I
let A := {c : ℝ | a c < 1}
let F := fun c => ENNReal.ofReal ((1/max (a c) (1/(2:ℝ)^N))^s)
have hcounts (j : ℕ) (hj : j ≤ 0+N) :
μ {c | a c < 1/(2:ℝ)^j} ≤ ENNReal.ofReal (C*(1/(2:ℝ)^j)^s) := by
rw [scalarCountingMeasure_apply]
have hh := ENNReal.ofReal_le_ofReal (hsmall j (by simpa using hj))
simpa only [ENNReal.ofReal_natCast,Set.mem_ofPred_eq] using hh
have hnear := truncated_scalar_potential_critical μ a ha ha0 hs 0 N hcounts
simp only [pow_zero,one_div_one,zero_add] at hnear
have hfar : (∫⁻ c in Aᶜ, F c ∂μ) ≤ (I.card:ℝ≥0∞) := by
calc
_ ≤ ∫⁻ _ in Aᶜ, (1:ℝ≥0∞) ∂μ := by
apply setLIntegral_mono' (by dsimp [A]; measurability)
intro c hc
have hc1 : 1 ≤ a c := le_of_not_gt hc
have hd : 1 ≤ max (a c) (1/(2:ℝ)^N) := hc1.trans (le_max_left _ _)
have hi : 1/max (a c) (1/(2:ℝ)^N) ≤ 1 :=
(div_le_one (zero_lt_one.trans_le hd)).2 hd
exact ENNReal.ofReal_le_one.mpr
((Real.rpow_le_rpow (by positivity) hi hs).trans_eq (Real.one_rpow s))
_ ≤ μ univ := by
simp only [lintegral_one,Measure.restrict_apply_univ]
exact measure_mono (subset_univ _)
_ = I.card := by rw [scalarCountingMeasure_apply]; simp
have htotal : (∑ c ∈ I, F c) ≤ (I.card:ℝ≥0∞)+
ENNReal.ofReal ((2:ℝ)^s)*(ENNReal.ofReal C+(N:ℝ≥0∞)*ENNReal.ofReal (C*(2:ℝ)^s)) := by
rw [← scalarCountingMeasure_lintegral]
change (∫⁻ c, F c ∂μ) ≤ _
rw [← lintegral_add_compl F (show MeasurableSet A from by dsimp [A]; measurability)]
exact (add_le_add hnear hfar).trans_eq (add_comm _ _)
change (∑ c ∈ I, ENNReal.ofReal ((1/max (a c) (1/(2:ℝ)^N))^s)) ≤ _ at htotal
rw [← ENNReal.ofReal_sum_of_nonneg (fun c _ => by positivity)] at htotal
have he : ENNReal.ofReal ((I.card:ℝ)+(2:ℝ)^s*(C+(N:ℝ)*(C*(2:ℝ)^s))) =
(I.card:ℝ≥0∞)+ENNReal.ofReal ((2:ℝ)^s)*
(ENNReal.ofReal C+(N:ℝ≥0∞)*ENNReal.ofReal (C*(2:ℝ)^s)) := by
rw [ENNReal.ofReal_add (Nat.cast_nonneg _) (by positivity),
ENNReal.ofReal_mul (by positivity),ENNReal.ofReal_add hC (by positivity),
ENNReal.ofReal_mul (Nat.cast_nonneg N)]
simp only [ENNReal.ofReal_natCast]
rw [← he] at htotal
exact (ENNReal.ofReal_le_ofReal_iff (by positivity)).mp htotal
end Falconer
end OAI
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