FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10

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import FormalConjecturesUtil

Bugeaud Collection of Conjectures and Open Questions: Spectrum of Sequence

References:

[Bug12] Bugeaud, Yann. "Distribution modulo one and Diophantine approximation." Vol. 193. Cambridge University Press, 2012. Chapter 10.

[Men73] Mendès France, Michel. "Les ensembles de Bésineau." Séminaire Delange-Pisot-Poitou 15.1 (1973): 1-6.


namespace Bugeaud04

The spectrum of a sequence $(x_n){n \ge 1}$ of real numbers is the set of irrational real numbers $\theta \in (0, 1)$ such that the sequence $(x_n - n\theta){n \ge 1}$ is not uniformly distributed modulo one.

def Spectrum (x : ℕ → ℝ) : Set ℝ :=
  {θ | θ ∈ Set.Ioo (0 : ℝ) 1 ∧ Irrational θ ∧
    ¬ IsEquidistributedModuloOne (fun n => x n - n * θ)}

Problem 10.4. Let $\xi$ be a non-zero real number and $\alpha > 1$ be a real number. The spectrum of the sequence $(\xi \alpha^n)_{n \ge 1}$ is at most countable. Posed by Mendès France [Men73].

@[category research open, AMS 11]
lemma declaration uses 'sorry'spectrum_xi_alpha_pow_countable (ξ : ℝ) (hξ : ξ ≠ 0) (α : ℝ) (hα : 1 < α) :
    (Spectrum (fun n => ξ * α ^ n)).Countable := byξ:ℝhξ:ξ ≠ 0α:ℝhα:1 < α⊢ (Spectrum fun n => ξ * α ^ n).Countable
  sorryAll goals completed! 🐙


end Bugeaud04