Bugeaud Collection of Conjectures and Open Questions: Confined Powers of Non-Pisot Numbers
Problem 10.7. Let be a positive real number. Are there arbitrarily large real numbers such that is not a Pisot number and all the fractional parts , , are lying in an interval of length $\varepsilon / ...
Mathematical statement
Problem 10.7. Let be a positive real number. Are there arbitrarily large real numbers such that is not a Pisot number and all the fractional parts , , are lying in an interval of length ? [Bug12b]
Statement source: Books statement material
Statement terms: Source-specific
Source-specific terms. Therefore does not assert reuse rights beyond attributed display.
Statement artifacts, not proofs
These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.
Pinned Lean formulation 1
problem_10_7
theorem problem_10_7 : answer(sorry) ↔ ∀ ε : ℝ, 0 < ε → ∀ M : ℝ, ∃ α : ℝ, M < α ∧ ¬ IsPisot α ∧ ∃ c : ℝ, ∀ n : ℕ, 1 ≤ n → Int.fract (α ^ n) ∈ Set.Icc c (c + ε / α) := by sorry- Statement source
- Formal Conjectures
- Lean version
- v4.27.0
- Placeholder
- Present; no proof artifact
- Source evidence
- Pinned source index
- Fidelity review
- Community formulation
References