Erdős Problem #1002 — open problem in number theory

Erdős Problem #1002

For any 0<α<1, let f(α,n)=1/log nΣ_1≤ k≤ n(1/2- α k). Does f(α,n) have an asymptotic distribution function? In other words, is there a non-decreasing function g such that g(-∞)=0, g(∞)=1, and lim_n→ ∞lvert α∈ (0,1): f(α,n)≤ crvert=g(c)?

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-1002,
  title        = {Erdős Problem #1002},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1002}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

For any $0 < α < 1$ , let $f(\alpha,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}- \{ \alpha k\}) $. D oes$ 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 $lim_{n \to \infty} ∣{α \in (0, 1) : f (α, n) \leq c}∣ = g (c)$ ?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1002». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

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))

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/1002. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1002
[Ke60] Kesten, Harry, Uniform distribution { $mod 1$ }. Ann. of Math. (2) (1960), 445--471.

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.