Bugeaud Collection of Conjectures and Open Questions: Confined Powers of Non-Pisot Numbers

Problem 10.7. Let ε be a positive real number. Are there arbitrarily large real numbers α such that α is not a Pisot number and all the fractional parts α^n, n ≥ 1, are lying in an interval of length ε / α? [Bug12b]

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-book-bugeaud-distribution-modulo-one-problem10-7,
  title        = {Bugeaud Collection of Conjectures and Open Questions: Confined Powers of Non-Pisot Numbers},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/book-bugeaud-distribution-modulo-one-problem10-7}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Problem 10.7. Let $ε$ be a positive real number. Are there arbitrarily large real numbers $α$ such that $α$ is not a Pisot number and all the fractional parts ${α^{n}}$ , $n \geq 1$ , are lying in an interval of length $ε / α$ ? [Bug12b]

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Books.BugeaudDistributionModuloOne.Problem10_7. answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

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)

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

[Bug12a] Bugeaud, Yann. "Distribution modulo one and Diophantine approximation." Vol. 193. Cambridge University Press, 2012. Chapter 10.
[Bug12b] Bugeaud, Yann, and Nikolay Moshchevitin. "On fractional parts of powers of real numbers close to 1." Mathematische Zeitschrift 271.3 (2012): 627-637.

Source and licence

Imported from Formal Conjectures (books), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.