Problem Statement
Let $A\subseteq \mathbb{N}$, and for each $n\in A$ choose some $X_n\subseteq \mathbb{Z}/n\mathbb{Z}$. Let\[B = \{ m\in \mathbb{N} : m\not\in X_n\pmod{n}\textrm{ for all }n\in A\textrm{ with }m>n\}.\]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?
Categories:
Number Theory Primitive Sets
Progress
Davenport and Erdős [DaEr36] proved that the answer is yes when $X_n=\{0\}$ for all $n\in A$. An alternative elementary proof was later given by Davenport and Erdős in [DaEr51].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 a generalisation of [25] (which is the case when $\lvert X_n\rvert=1$ for all $n\in A$).
Source: erdosproblems.com/486 | Last verified: January 15, 2026