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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 | import OAI.MeasureTheory.Falconer.Geometry.ProjectionBuffer
import OAI.MeasureTheory.Falconer.Estimates.RescaledConditional
namespace OAI
/-! # Projection buffers for the full original conditional laws -/
namespace Falconer.Regularization
open Set MeasureTheory
variable {d : ℕ}
theorem conditional_pair_diameter (Q : DyadicCube d) (hQ : 0 < Q.side)
(ρ : Measure (Space d)) [IsFiniteMeasure ρ] (P : CellPath d) :
∀ᵐ z : Space d × Space d ∂
(ProbabilityTheory.cond ρ (Q.descendant P).carrier).prod
(ProbabilityTheory.cond ρ (Q.descendant P).carrier),
‖z.1-z.2‖ ≤ Real.sqrt d*(Q.side/(2:ℝ)^P.length) := by
have hs : ∀ᵐ z ∂ProbabilityTheory.cond ρ (Q.descendant P).carrier,
z ∈ (Q.descendant P).carrier :=
ProbabilityTheory.ae_cond_of_forall_mem (Q.descendant P).measurable (fun _ hz => hz)
apply (Measure.ae_prod_iff_ae_ae (by measurability)).mpr
filter_upwards [hs] with x hx
filter_upwards [hs] with y hy
have hh := (Q.descendant P).dist_le (by rw [Q.descendant_side]; positivity) hx hy
simpa only [dist_eq_norm,Q.descendant_side] using hh
theorem conditional_projection_buffer (Q : DyadicCube d) (hQ : Q.side = 1)
(ρ : Measure (Space d)) [IsFiniteMeasure ρ] (P : CellPath d)
{δ W W' T : ℝ} (hδ : 0 ≤ δ) {v w : Sphere d} (hvw : dist v w ≤ δ)
(hwidth : W+(Real.sqrt d/(2:ℝ)^P.length)*δ ≤ W')
(hw : w ∈ projectionGoodSet (ProbabilityTheory.cond ρ (Q.descendant P).carrier) W' T) :
v ∈ projectionGoodSet (ProbabilityTheory.cond ρ (Q.descendant P).carrier) W T := by
apply projectionGoodSet_buffer _ (by positivity) hδ _ hvw hwidth hw
simpa only [hQ,mul_one_div] using conditional_pair_diameter Q (by rw [hQ]; norm_num) ρ P
end Falconer.Regularization
end OAI
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