Erdős Problem #486

Let \(A \subseteq \mathbb{N}\), and for each \(n \in A\) choose some \(X_n \subseteq \mathbb{Z}/n\mathbb{Z}\). Let \(B = \{m \in \mathbb{N} : m \not\in X_n \pmod{n} \text{ for all } n \in A \text{ with } m > n\}\). Must \(B\) have a logarithmic density? The set \(A\) is encoded by taking \(X_n = \emptyset\) for \(n \notin A\). Only positive moduli \(n\) are considered, since \(\mathbb{Z}/0\mathbb{Z} = \mathbb{Z}\) would allow \(B\) to be an arbitrary set. The answer is no: Wang [Wa26] (with GPT-5.6) constructed a congruence system whose survivor set \(B\) has lower logarithmic density at most \(177/200\) and upper logarithmic density at least \(49/50\). The first linked formal proof establishes this for the real-cutoff normalisation \(\frac{1}{\log x}\sum_{m < x, m \in B} \frac{1}{m}\) with the moduli restricted to \(A\); the second derives the statement below (`Set.HasLogDensity`, all moduli) from it.