| 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 |
|
|
