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9eeb9c7 | 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 | 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
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