| import OAI.Analysis.IrrationalRotation.Model.Analytic |
| import OAI.Analysis.IrrationalRotation.Model.ContinuityCoboundary |
| import Mathlib |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped ENNReal Topology |
|
|
| namespace Invariant233 |
|
|
| structure BackgroundBumpData (θ : ℝ) where |
| radius : ℕ → ℝ |
| bump : ℕ → Circle → ℝ |
| radius_pos : ∀ n, 0 < radius n |
| radius_lt : ∀ n, radius n < 1 / ((n : ℝ) + 1) |
| continuous_bump : ∀ n, Continuous (bump n) |
| bump_zero : ∀ n, bump n 0 = 1 |
| bump_bounds : ∀ n x, 0 ≤ bump n x ∧ bump n x ≤ 1 |
| bump_support : ∀ n, Function.support (bump n) ⊆ Metric.closedBall 0 (radius n) |
| disjoint_translates : ∀ n, Pairwise (fun i j : Fin (2 ^ (n + 2)) => |
| Disjoint |
| ((fun x : Circle => x + ((i : ℕ) + 1) • (θ : Circle)) '' |
| Metric.closedBall 0 (radius n)) |
| ((fun x : Circle => x + ((j : ℕ) + 1) • (θ : Circle)) '' |
| Metric.closedBall 0 (radius n))) |
| weighted_support_bound : |
| (∑' n, (2 : ℝ≥0∞) ^ (n + 2) * circleMeasure (Metric.closedBall 0 (radius n))) ≤ 1 / 2 |
|
|
| def backgroundNegative {θ : ℝ} (d : BackgroundBumpData θ) (n : ℕ) (x : Circle) : ℝ := |
| InvariantProjection.Continuity.backwardAverage (Equiv.addRight (θ : Circle)) |
| (d.bump n) (2 ^ (n + 2)) x |
|
|
| def backgroundCost {θ : ℝ} (d : BackgroundBumpData θ) (x : Circle) : ℝ := |
| 1 + (∑' n, d.bump n x) - (∑' n, backgroundNegative d n x) |
|
|
| def backgroundTransferSum {θ : ℝ} (d : BackgroundBumpData θ) (x : Circle) : ℝ := |
| ∑' n, InvariantProjection.Continuity.backgroundTransfer (Equiv.addRight (θ : Circle)) |
| (d.bump n) (2 ^ (n + 2)) x |
|
|
| end Invariant233 |
|
|
| end |
|
|
| end OAI |
|
|
