| 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 |
|
|
| / |
| 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) |
|
|
| / |
| 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 |
|
|
