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

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.

theorem Bugeaud07.problem_10_7 : True ↔ ∀ (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)

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