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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 statement2 of 5

The study of ϕA\phi_A was introduced by Cassels [Ca50b], who proved that there exist sequences such that lim inf⁡N→∞1N∑k≤NϕA(k)nk=0. \liminf_{N\to \infty}\frac{1}{N}\sum_{k\leq N}\frac{\phi_A(k)}{n_k}=0.

FormalConjectures/ErdosProblems/1000.leanErdos1000.erdos_1000.variants.liminf_eq_zero1 lineExact file
∃ n, StrictMono n ∧ 0 < n 0 ∧ Filter.liminf (Erdos1000.phiAvg n) Filter.atTop = 0
SolvedStatement only, no proof