Mahler32.lean`

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

4 declarations, 3 contain sorry, 0 declared as axioms

Mahler's 3/2 Problem

Reference: Wikipedia

Imports (1)

FormalConjecturesUtil

Imported by 2 files

def Mahler32.Ω Documented declaration: Ω line 36 source
Source documentation
For a real number α, define Ω(α) as $$ \Omega (\alpha )=\inf _{\theta > 0}\left({\limsup _{n\rightarrow \infty }\left\lbrace {\theta \alpha ^{n}}\right\rbrace -\liminf _{n\rightarrow \infty }\left\lbrace {\theta \alpha ^{n}}\right\rbrace }\right). $$

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: InfSet.sInf, Set.ofPred, Exists, LT.lt, OfNat.ofNat, Eq, HSub.hSub, Filter.limsup, Nat, Int.fract, HMul.hMul, HPow.hPow, Filter.atTop, Filter.liminf
18 instances, coercions and auxiliary constants left out.
```
noncomputable def Ω (α : ℝ) : ℝ :=
…
```
theorem Mahler32.mahler_conjecture The Mahler Conjecture states that there are no Z-numbers sorry line 42 source
Source documentation
The Mahler Conjecture states that there are no Z-numbers.

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 mahler_conjecture (x : ℝ) (hx : IsZNumber x) : False := by
…
```
Mentions: IsZNumber
theorem Mahler32.mahler_conjecture.variants.consequence Mahler's conjecture would follow if Ω(3/2) exceeded 1/2: a Z-number x has all fractional parts {x (3/2)^n} below 1/2, so Ω(3/2) ≤ 1/2 sorry line 48 source
Source documentation
Mahler's conjecture would follow if Ω(3/2) exceeded 1/2: a Z-number x has all fractional parts {x (3/2)^n} below 1/2, so Ω(3/2) ≤ 1/2.

depends on sorry what this means
Statement uses: LT.lt, Real, HDiv.hDiv, OfNat.ofNat, Nat, Mahler32.Ω, IsZNumber, False
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem mahler_conjecture.variants.consequence (H : 1 / 2 < Ω (3 / 2)) :
    type_of% mahler_conjecture := by
…
```
Mentions: Mahler32.Ω, Mahler32.mahler_conjecture
theorem Mahler32.mahler_conjecture.variants.flatto_lagarias_pollington Documented declaration: flatto_lagarias_pollington sorry line 56 source
Source documentation
Flatto, Lagarias and Pollington proved that for all rational p/q > 1 in lowest terms with q ≥ 2, we have Ω(p/q) ≥ 1/p: for every θ > 0, the limit points of {θ (p/q)^n} are not contained in any interval of length less than 1/p.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Nat.Coprime, LE.le, Real, HDiv.hDiv, Mahler32.Ω
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem mahler_conjecture.variants.flatto_lagarias_pollington (p q : ℕ) (hq : 1 < q)
    (hpq : p.Coprime q) (hpq' : q < p) : 1 / p ≤ Ω (p / q) := by
…
```
Mentions: Mahler32.Ω