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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 | import OAI.MeasureTheory.Falconer.Shells.FrequencyShell
namespace OAI
/-! # Covering the actual radial interval by its dyadic frequency annuli -/
namespace Falconer
open Set MeasureTheory
open scoped ENNReal
variable {d : ℕ}
noncomputable def radialIntervalMeasure (ν : Measure (Sphere d)) (H : ℝ) : Measure (Space d) :=
Measure.map radialFrequencyMap ((volume.restrict (Icc (-H) H)).prod ν)
instance radialIntervalMeasure_finite (ν : Measure (Sphere d)) [IsFiniteMeasure ν] (H : ℝ) :
IsFiniteMeasure (radialIntervalMeasure ν H) := by
unfold radialIntervalMeasure
infer_instance
theorem radial_interval_cover_step (n : ℕ) :
Icc (-((2:ℝ)^(n+1))) ((2:ℝ)^(n+1)) ⊆
Icc (-((2:ℝ)^n)) ((2:ℝ)^n) ∪ radialFrequencyAnnulus ((2:ℝ)^n) := by
intro t ht
have hh : |t| ≤ (2:ℝ)^(n+1) := abs_le.mpr ht
by_cases hs : |t| ≤ (2:ℝ)^n
· exact Or.inl (abs_le.mp hs)
· right
refine ⟨(le_of_not_ge hs),?_⟩
simpa only [pow_succ,mul_comm] using hh
theorem radial_interval_measure_step (ν : Measure (Sphere d)) [IsFiniteMeasure ν] (n : ℕ) :
radialIntervalMeasure ν ((2:ℝ)^(n+1)) ≤
radialIntervalMeasure ν ((2:ℝ)^n)+radialFrequencyMeasure ν ((2:ℝ)^n) := by
have hvol : volume.restrict (Icc (-((2:ℝ)^(n+1))) ((2:ℝ)^(n+1))) ≤
volume.restrict (Icc (-((2:ℝ)^n)) ((2:ℝ)^n))+
volume.restrict (radialFrequencyAnnulus ((2:ℝ)^n)) :=
(Measure.restrict_mono (radial_interval_cover_step n) le_rfl).trans (Measure.restrict_union_le _ _)
have hm : Measurable (@radialFrequencyMap d) := by unfold radialFrequencyMap; fun_prop
have h := Measure.map_mono (Measure.prod_mono hvol (le_refl ν)) hm
simpa only [radialIntervalMeasure,radialFrequencyMeasure,Measure.add_prod,
Measure.map_add _ _ hm] using h
theorem radial_interval_dyadic_cover (ν : Measure (Sphere d)) [IsFiniteMeasure ν] (n : ℕ) :
radialIntervalMeasure ν ((2:ℝ)^n) ≤ radialIntervalMeasure ν 1+
∑ j ∈ Finset.range n, radialFrequencyMeasure ν ((2:ℝ)^j) := by
induction n with
| zero => simp
| succ n ih =>
calc
_ ≤ radialIntervalMeasure ν ((2:ℝ)^n)+radialFrequencyMeasure ν ((2:ℝ)^n) :=
radial_interval_measure_step ν n
_ ≤ (radialIntervalMeasure ν 1+∑ j ∈ Finset.range n, radialFrequencyMeasure ν ((2:ℝ)^j))+
radialFrequencyMeasure ν ((2:ℝ)^n) := add_le_add ih le_rfl
_ = _ := by rw [Finset.sum_range_succ,add_assoc]
end Falconer
end OAI
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