Source labels openChecked July 26, 2026

Littlewood conjectures

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.

Number theory AMS 11

Mathematical statement

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.

Statement source: Wikipedia statement material

Statement terms: CC-BY-SA-4.0

Attributed source material. Reuse must follow the linked attribution and share-alike terms.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

littlewood_conjecture

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem littlewood_conjecture (α β : ℝ) :    atTop.liminf (fun (n : ℕ) ↦ n * distToNearestInt (n * α) * distToNearestInt (n * β)) = 0 := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization Wikipediabackground