Let $α, β \in R$ . Is it true that $n \to \infty lim inf n ∥ n α ∥∥ n β ∥ = 0$ where $∥ x ∥$ is the distance from $x$ to the nearest integer?

Worked, still open.

diophantine approximation · open · formalized (Lean) · 0 attempts

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vela registry pull vfr_37aec80d874a0239

vela reproduce examples/erdos-problems

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unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

What you wrote is **exactly Littlewood’s conjecture**:

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

AMS 11 · open (literature)

theorem erdos_495 : answer(sorry) ↔ ∀ α β : ℝ, liminf (fun n : ℕ ↦ (n : ℝ) * distToNearestInt (n * α)
  * distToNearestInt (n * β)) atTop = 0

open