Problem10_9.lean`

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4 declarations, 3 contain sorry, 0 declared as axioms

Bugeaud Collection of Conjectures and Open Questions: Mahler's Z-numbers

See also FormalConjectures/Wikipedia/Mahler32.lean.

References:

[Bug12] Bugeaud, Yann. "Distribution modulo one and Diophantine approximation." Vol. 193. Cambridge University Press, 2012. Chapter 10.
[Mah68] Mahler, Kurt. "An unsolved problem on the powers of 3/2." Journal of the Australian Mathematical Society 8.2 (1968): 313-321.
[FLP95] Flatto, Leopold, Jeffrey C. Lagarias, and Andrew D. Pollington. "On the range of fractional parts $\{\xi(p/q)^n\}$." Acta Arithmetica 70.2 (1995): 125-147.

Imports (2)

theorem Bugeaud09.problem_10_9 Problem 10.9 sorry line 44 source
Source documentation
Problem 10.9. There are no real numbers $\xi$ such that $0 \le \{\xi (3/2)^n\} < 1/2$ for every positive integer $n$, i.e. no Z-number exists. Posed by Mahler [Mah68].

depends on sorry what this means
Statement uses: Real, IsZNumber, False
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem problem_10_9 : type_of% Mahler32.mahler_conjecture := by
…
```
Mentions: Mahler32.mahler_conjecture
theorem Bugeaud09.not_isZNumber_one Sanity check: 1 is not a Z-number, since for n = 1 we have $\{1 \cdot (3/2)^1\} = \{3/2\} = 1/2 \not< 1/2$ line 52 source
Source documentation
Sanity check: 1 is not a Z-number, since for n = 1 we have $\{1 \cdot (3/2)^1\} = \{3/2\} = 1/2 \not< 1/2$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Not, IsZNumber, OfNat.ofNat, Real
Proof uses: LT.lt, Nat, LE.le, Int.fract, HMul.hMul, HPow.hPow, HDiv.hDiv, False, le_rfl, Eq, Eq.mpr, HSub.hSub, Int.floor, id, congrArg, Int.fract.eq_1, Mathlib.Meta.NormNum.isNNRat_eq_true, Mathlib.Meta.NormNum.IsRat.to_isNNRat, Mathlib.Meta.NormNum.isRat_sub, Int.ofNat and 18 more
64 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
```
theorem not_isZNumber_one : ¬ IsZNumber 1 := by
…
```
Mentions: IsZNumber
theorem Bugeaud09.problem_10_9.variants.measure_zero Mahler [Mah68] proved that the set of Z-numbers has Lebesgue measure zero sorry line 64 source
Source documentation
Mahler [Mah68] proved that the set of Z-numbers has Lebesgue measure zero.

depends on sorry what this means
Statement uses: Eq, ENNReal, MeasureTheory.Measure, Real, Set, MeasureTheory.MeasureSpace.volume, Set.ofPred, IsZNumber, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem problem_10_9.variants.measure_zero :
    MeasureTheory.volume {ξ : ℝ | IsZNumber ξ} = 0 := by
…
```
Mentions: IsZNumber
theorem Bugeaud09.problem_10_9.variants.dimH_lt_one Flatto, Lagarias, and Pollington [FLP95] proved that the set of Z-numbers has Hausdorff dimension strictly less than $1$ sorry line 73 source
Source documentation
Flatto, Lagarias, and Pollington [FLP95] proved that the set of Z-numbers has Hausdorff dimension strictly less than $1$.

depends on sorry what this means
Statement uses: LT.lt, ENNReal, dimH, Real, Set.ofPred, IsZNumber, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem problem_10_9.variants.dimH_lt_one :
    dimH {ξ : ℝ | IsZNumber ξ} < 1 := by
…
```
Mentions: IsZNumber