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Erdős problem 1000

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Let A={n1<n2<⋯ }A=\{n_1<n_2<\cdots\} be an infinite sequence of integers, and let ϕA(k)\phi_A(k) count the number of 1≤m≤nk1\leq m\leq n_k such that the fraction mnk\frac{m}{n_k} does not have denominator njn_j for j<kj<k when written in lowest form; equivalently, nk(m,nk)≠nj \frac{n_k}{(m,n_k)}\neq n_j for all 1≤j<k1\leq j<k.

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FormalConjectures/ErdosProblems/

1000.lean

Retained formal statement3 of 5

In fact he proved that if lim inf⁡ϕA(k)nk=0\liminf \frac{\phi_A(k)}{n_k}=0 then lim sup⁡ϕA(k)nk=1\limsup \frac{\phi_A(k)}{n_k}=1.

FormalConjectures/ErdosProblems/1000.leanErdos1000.erdos_1000.variants.limsup_eq_one5 linesExact file
∀ (n : ℕ → ℕ),  StrictMono n →    0 < n 0 →      Filter.liminf (fun k => ↑(Erdos1000.phiSeq n k) / ↑(n k)) Filter.atTop = 0 →        Filter.limsup (fun k => ↑(Erdos1000.phiSeq n k) / ↑(n k)) Filter.atTop = 1
SolvedStatement only, no proof