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Erdős problem 1002

For any 0<α<10<\alpha<1, let f(α,n)=1log⁡n∑1≤k≤n(12−{αk})f(\alpha,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}- \{ \alpha k\}). Does f(α,n)f(\alpha,n) have an asymptotic distribution function?

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No reviewed Result is current in Vela Mathematics Program. Retained source material is shown below.

Retained declaration

FormalConjectures/ErdosProblems/1002.lean

Formal Conjectures

FormalConjectures/ErdosProblems/1002.leanErdos1002.erdos_100214 linesExact file
sorry ↔  ∃ g,    Monotone g ∧      Filter.Tendsto g Filter.atBot (nhds 0) ∧        Filter.Tendsto g Filter.atTop (nhds 1) ∧          ∀ (c : ℝ),            Filter.Tendsto              (fun n =>                (MeasureTheory.volume                    {α |                      α ∈ Set.Ioo 0 1 ∧                        (fun α n => 1 / Real.log ↑n * ∑ k ∈ Finset.Icc 1 n, (1 / 2 - Int.fract (α * ↑k))) α n ≤                          c}).toReal)              Filter.atTop (nhds (g c))
OpenStatement only, no proof

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