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caec230 | 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 | import Mathlib.Topology.Order.Monotone
import Mathlib.Topology.Instances.Real.Lemmas
import Mathlib.Analysis.SpecialFunctions.Pow.Real
namespace OAI
/-!
# Endpoint of the thin-tube bootstrap
This is the order-theoretic step in Proposition `prep:thin-upgrade`.
Its hypotheses describe the closed set of available exponents inside
`[k - 1, k - 1 / 2]`, with strict improvement below the endpoint.
-/
namespace Falconer
open Set
/-- A closed family of available exponents containing the integer starting
exponent and admitting a strict improvement below the desired endpoint
contains the half-integer endpoint. -/
theorem thin_tube_bootstrap_endpoint (k : ℕ) (A : Set ℝ)
(hclosed : IsClosed A) (hstart : (k : ℝ) - 1 ∈ A)
(hupper : ∀ q ∈ A, q ≤ (k : ℝ) - 1 / 2)
(himprove : ∀ q ∈ A, q < (k : ℝ) - 1 / 2 → ∃ q' ∈ A, q < q') :
(k : ℝ) - 1 / 2 ∈ A := by
have hnonempty : A.Nonempty := ⟨_, hstart⟩
have hbounded : BddAbove A := ⟨(k : ℝ) - 1 / 2, hupper⟩
have hsup : sSup A ∈ A := hclosed.csSup_mem hnonempty hbounded
have hsup_upper : sSup A ≤ (k : ℝ) - 1 / 2 := hupper _ hsup
by_cases heq : sSup A = (k : ℝ) - 1 / 2
· exact heq ▸ hsup
· obtain ⟨q, hq, hlt⟩ := himprove _ hsup (lt_of_le_of_ne hsup_upper heq)
exact False.elim ((not_lt_of_ge (le_csSup hbounded hq)) hlt)
/-- At resolution `R⁻¹`, increasing the cap exponent by at most `e` costs
only the allowed factor `R ^ e`. This is the quantitative inequality used
when closing the set of available exponents in `prep:thin-upgrade`. -/
theorem cap_exponent_loss_transfer {R u q q' e : ℝ}
(hR : 1 ≤ R) (hu : R⁻¹ ≤ u) (hqq : q ≤ q') (hgap : q' - q ≤ e) :
u ^ q ≤ R ^ e * u ^ q' := by
have hRpos : 0 < R := lt_of_lt_of_le zero_lt_one hR
have hupos : 0 < u := (inv_pos.mpr hRpos).trans_le hu
have hcompare : u ^ (q - q') ≤ R ^ e := by
calc
u ^ (q - q') ≤ (R⁻¹) ^ (q - q') :=
Real.rpow_le_rpow_of_nonpos (inv_pos.mpr hRpos) hu (sub_nonpos.mpr hqq)
_ = R ^ (q' - q) := by
rw [← Real.rpow_neg_eq_inv_rpow]
congr 1
ring
_ ≤ R ^ e := Real.rpow_le_rpow_of_exponent_le hR hgap
calc
u ^ q = u ^ (q - q') * u ^ q' := by
rw [← Real.rpow_add hupos]
congr 1
ring
_ ≤ R ^ e * u ^ q' := mul_le_mul_of_nonneg_right hcompare (Real.rpow_nonneg hupos.le _)
end Falconer
end OAI
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