Source labels openChecked July 26, 2026

Bugeaud Collection of Conjectures and Open Questions: Fractional Parts of Powers

Problem 10.1. Are there a transcendental number $\alpha$ and a positive real number $\xi$ such that $\lVert \xi \alpha^n \rVert$ tends to~ $0$ as~ $n$ tends to infinity? [Har19] (Trivial for $|\alpha| < 1$ )

Number theory AMS 11

Mathematical statement

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_1

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem problem_10_1 : answer(sorry) ↔    ∃ (α ξ : ℝ), 1 < |α| ∧ Transcendental ℚ α ∧ 0 < ξ ∧      Filter.Tendsto (fun n : ℕ ↦ distToNearestInt (ξ * α ^ n)) Filter.atTop (nhds 0) := 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

Pinned Formal Conjectures sourceformalization Wal03background