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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 48 49 50 | import OAI.Analysis.IrrationalRotation.Model.ContinuousCounterexample
import OAI.Analysis.IrrationalRotation.Rotation.CircleParametersExistence
import OAI.Analysis.IrrationalRotation.Rotation.CircleMeasurableObstruction
import OAI.Analysis.IrrationalRotation.Analysis.ContinuousCounterexampleAssembly
import Mathlib.Analysis.Normed.Algebra.GelfandFormula
namespace OAI
/-! The spectral-radius hypothesis and its consequences for prescribed irrational angles. -/
noncomputable section
open Filter
open scoped ENNReal Topology
namespace Invariant233
structure PublishedRotationSpectralInput (θ : ℝ) : Prop where
radius_zero : ∀ f : C(Circle, ℝ), IsContinuousCircleWeight f →
spectralRadius ℂ (Rotation.weightedReal θ f) = 0
theorem PublishedRotationSpectralInput.normQuasinilpotent
{θ : ℝ} (hpub : PublishedRotationSpectralInput θ)
(f : C(Circle, ℝ)) (hf : IsContinuousCircleWeight f) :
NormQuasinilpotent (Rotation.weightedReal θ f) := by
have h := spectrum.pow_norm_pow_one_div_tendsto_nhds_spectralRadius
(Rotation.weightedReal θ f)
rw [hpub.radius_zero f hf] at h
have ht := (ENNReal.tendsto_toReal (by simp : (0 : ℝ≥0∞) ≠ ∞)).comp h
simpa only [NormQuasinilpotent, Function.comp_def, ENNReal.toReal_zero,
ENNReal.toReal_ofReal (Real.rpow_nonneg (norm_nonneg _) _)] using ht
namespace Results
theorem prescribed_irrational_counterexample_of_published_spectral_input
(θ : ℝ) (hθ : Irrational θ) (hpub : PublishedRotationSpectralInput θ) :
∃ f : C(Circle, ℝ), Rotation.IsContinuousCounterexample θ f := by
obtain ⟨p, hp⟩ := exists_circle_parameters hθ
obtain ⟨f, hf, hmem, hinv⟩ :=
hp.exists_continuous_invariant_weight hθ
(hp.measurableOperator_hasOnlyTrivialInvariantProjections hθ)
exact ⟨f, hf, hmem, hinv, Rotation.weightedReal_ne_zero_of_isContinuousCircleWeight θ f hf,
hpub.normQuasinilpotent f hf, Rotation.weightedReal_vectorDeterminant_zero θ f hf⟩
end Results
end Invariant233
end
end OAI
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