Problem10_7.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

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.

Imports (1)

FormalConjecturesUtil

theorem Bugeaud07.problem_10_7 Problem 10.7 sorry line 41 source
Source documentation
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]

depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Exists, And, Not, IsPisot, Nat, LE.le, Membership.mem, Set, Set.Icc, HAdd.hAdd, HDiv.hDiv, Int.fract, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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
…
```
Mentions: IsPisot, c