OPEN
Let $A\subseteq \mathbb{N}$, and for each $n\in A$ choose some $X_n\subseteq \mathbb{Z}/n\mathbb{Z}$. Let
\[B = \{ m\in \mathbb{N} : m\not\in X_n\pmod{n}\textrm{ for all }n\in A\}.\]
Must $B$ have a logarithmic density, i.e. is it true that
\[\lim_{x\to \infty} \frac{1}{\log x}\sum_{\substack{m\in B\\ m<x}}\frac{1}{m}\]
exists?
Davenport and Erdős
[DaEr37] proved that the answer is yes when $X_n=\{0\}$ for all $n\in A$. The problem considers logarithmic density since Besicovitch
[Be34] showed examples exist without a natural density, even when $X_n=\{0\}$ for all $n\in A$.
This is very similar to [25].
