Source labels openChecked July 26, 2026

Littlewood conjectures

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.

Number theory AMS 11

Mathematical statement

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.

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

padic_littlewood_conjecture

Canonical source

Complete statement target, proof intentionally absentLean 4

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  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