LittlewoodConjecture.lean`

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

2 declarations, 2 contain sorry, 0 declared as axioms

Littlewood conjectures

References:

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

theorem LittlewoodConjecture.littlewood_conjecture For any two real numbers sorry line 46 source
Source documentation
For any two real numbers $\alpha$ and $\beta$, $$ \liminf_{n\to\infty} n\||n\alpha\||\||n\beta\|| = 0 $$ where $\||x\|| := \min(|x - \lfloor x \rfloor|, |x - \lceil x \rceil|)$ is the distance to the nearest integer.

depends on sorry what this means
Statement uses: Real, Eq, Filter.liminf, Nat, HMul.hMul, distToNearestInt, Filter.atTop, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
19 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem littlewood_conjecture (α β : ℝ) :
    atTop.liminf (fun (n : ℕ) ↦ n * distToNearestInt (n * α) * distToNearestInt (n * β)) = 0 := by
…
```
Mentions: distToNearestInt
theorem LittlewoodConjecture.padic_littlewood_conjecture For real number sorry line 59 source
Source documentation
For real number $\alpha$ and prime $p$, $$ \liminf\_{n \to\infty} n |n|\_{p}\||n\alpha\|| = 0 $$ where $\||x\|| := \min(|x - \lfloor x \rfloor|, |x - \lceil x \rceil|)$ is the distance to the nearest integer, and $|x|\_{p}$ is the $p$-adic norm.

depends on sorry what this means
Statement uses: Real, Nat, Nat.Prime, Eq, Filter.liminf, HMul.hMul, padicNorm, Rat, distToNearestInt, Filter.atTop, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
25 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem padic_littlewood_conjecture (α : ℝ) (p : ℕ) (hp : p.Prime) :
    atTop.liminf (fun (n : ℕ) ↦ n * padicNorm p n * distToNearestInt (n * α)) = 0 := by
…
```
Mentions: distToNearestInt