Bugeaud Collection of Conjectures and Open Questions: Spectrum of Sequence
Bugeaud Collection of Conjectures and Open Questions: Spectrum of Sequence
Inferred from the community field; finer classification has not been reviewed.
Problem statement
spectrum_xi_alpha_pow_countable
Problem 10.4. Let be a non-zero real number and be a real number. The spectrum of the sequence is at most countable. Posed by Mendès France [Men73].
Lean 4
Formal statement
Lean 4/-
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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 spectrum_xi_alpha_pow_countable (ξ : ℝ) (hξ : ξ ≠ 0) (α : ℝ) (hα : 1 < α) :
(Spectrum (fun n => ξ * α ^ n)).Countable := by
sorry
end Bugeaud04
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Background
Source snapshot inspected on 10 September 2026. Subsequent research status and the accuracy of source formalizations have not been independently certified.
Fixed commit b82b08faa9006484021c12005ab41287fb2ffb69. This file contains 1 source-labeled research-open declarations. Other declarations in the attached file can be definitions, solved results or tests. The file is preserved as source material; it has not been compiled or proved by this import.
Copyright The Formal Conjectures Authors. Code: Apache-2.0; other original material: CC BY 4.0. Third-party sources retain their own licenses. Original source files and notices preserved.
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