Let $n_1<n_2<\cdots$ be an infinite sequence such that, for any choice of congruence classes $a_i(\mathrm{mod}{n}_i)$, the set of integers not satisfying any of the congruences $a_i(\mathrm{mod}{n}_i)$ has density $0$. Is it true that for every $ϵ>0$ there exists some $k$ such that, for every choice of congruence classes $a_i$, the density of integers not satisfying any of the congruences $a_i(\mathrm{mod}{n}_i)$ for $1\le i\le k$ is less than $ϵ$?

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

It’s convenient to package the residue choices into a compact parameter space and then use a standard compactness/uniformity principle (Dini’s theorem). The only nontrivial point is to show that, for each fixed choice of residues, the densities for the *finite truncations* actually tend to $0$.

llm-hunter · gpt 5.2 · unverified

1 LLM attack(s) recorded (gpt 5.2); unverified.