Source labels openChecked July 26, 2026

Erdős Problem 1002

For any $0<\alpha<1$ , let $f(\alpha,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}- \{ \alpha k\})$ . Does $f(\alpha,n)$ have an asymptotic distribution function?

Number theory AMS 11

Mathematical statement

For any $0<\alpha<1$ , let $f(\alpha,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}- \{ \alpha k\})$ . Does $f(\alpha,n)$ have an asymptotic distribution function?

In other words, is there a non-decreasing function $g$ such that $g(-\infty)=0$ , $g(\infty)=1$ , and $\lim_{n\to \infty}\lvert \{ \alpha\in (0,1): f(\alpha,n)\leq c\}\rvert=g(c)$ ?

Statement source: Erdős Problems statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

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

erdos_1002

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem erdos_1002 :    answer(sorry) ↔      ∃ g : ℝ → ℝ, Monotone g ∧      Tendsto g atBot (𝓝 0) ∧      Tendsto g atTop (𝓝 1) ∧      letI f :=  fun (α : ℝ) (n : ℕ) ↦        (1 / log n) * ∑ k ∈ Icc (1 : ℕ) n, (1 / 2 - Int.fract (α * k))      ∀ c : ℝ, Tendsto (fun (n : ℕ) ↦ (volume { α | α ∈ Ioo (0 : ℝ) 1 ∧ f α n ≤ c }).toReal)        atTop (𝓝 (g c)) := 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 erdosproblems.com/1002background