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All Random Solved Random Open
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 $n\in A$.