File size: 2,273 Bytes
6aac220 | 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 | import OAI.GroupTheory.PolycyclicRecognition.Analysis.BoundedPathGrowth
import OAI.GroupTheory.PolycyclicRecognition.Analysis.MeasurePacking
namespace OAI
/-!
# From polynomial-volume thickenings to bounded-height path packing
The endpoint set need not be measurable. We pack finite separated subsets using
Haar mass and then take a maximal one. Only the enclosing coordinate regions
are measured; this is the precise bridge needed from the Lie-coordinate bounds.
-/
namespace PolycyclicRecognition
open MeasureTheory MeasureTheory.Measure Metric
open scoped ENNReal
variable {G W : Type*} [Group G] [MetricSpace G] [MeasurableSpace G]
[BorelSpace G] [MeasurableMul G] [IsIsometricSMul G G] [NormedAddCommGroup W]
/-- Endpoints of at most `n` bounded-height steps from a fixed starting point. -/
def boundedChainEndpoints (π : G → W) (L M : ℝ) (n : ℕ) (x : G) : Set G :=
{y | ∃ m ≤ n, BoundedHeightChain π L M m x y}
/-- A polynomial-volume region containing the half-unit thickening of the
bounded-chain endpoints gives the required polynomial covering bound. -/
theorem boundedChain_cover_of_volume
(μ : Measure G) [IsMulLeftInvariant μ] (π : G → W)
(x : G) (L M : ℝ) (C b : ℕ) (region : ℕ → Set G)
(hq0 : μ (ball (1 : G) (1 / 2)) ≠ 0)
(hqtop : μ (ball (1 : G) (1 / 2)) ≠ ⊤)
(hthick : ∀ n y, y ∈ boundedChainEndpoints π L M n x →
ball y (1 / 2) ⊆ region n)
(hvol : ∀ n, μ (region n) ≤
((C * (1 + n) ^ b : ℕ) : ℝ≥0∞) * μ (ball (1 : G) (1 / 2))) :
∃ centers : ℕ → Finset G,
(∀ n, (centers n).card ≤ C * (1 + n) ^ b) ∧
∀ n m, m ≤ n → ∀ y, BoundedHeightChain π L M m x y →
∃ c ∈ centers n, dist y c ≤ 1 := by
classical
have hn (n : ℕ) := finite_cover_set_of_separated_bound
(boundedChainEndpoints π L M n x) (r := 1) (by norm_num) (C * (1 + n) ^ b)
(fun t ht hs => separated_card_le_polynomial μ t (1 / 2) (region n) C b n hq0 hqtop
(by intro a ha d hd had; have := hs a ha d hd had; linarith)
(fun y hy => hthick n y (ht hy)) (hvol n))
choose centers _ hcard hcover using hn
exact ⟨centers, hcard, fun n m hm y hy => hcover n y ⟨m, hm, hy⟩⟩
end PolycyclicRecognition
end OAI
|