Problem Statement
Let $n_1<n_2<\cdots$ be an infinite sequence such that, for any choice of congruence classes $a_i\pmod{n_i}$, the set of integers not satisfying any of the congruences $a_i\pmod{n_i}$ has density $0$.
Is it true that for every $\epsilon>0$ there exists some $k$ such that, for every choice of congruence classes $a_i$, the density of integers not satisfying any of the congruences $a_i\pmod{n_i}$ for $1\leq i\leq k$ is less than $\epsilon$?
Is it true that for every $\epsilon>0$ there exists some $k$ such that, for every choice of congruence classes $a_i$, the density of integers not satisfying any of the congruences $a_i\pmod{n_i}$ for $1\leq i\leq k$ is less than $\epsilon$?
Categories:
Number Theory Covering Systems
Progress
The latter condition is clearly sufficient, the problem is if it's also necessary. The assumption implies $\sum \frac{1}{n_i}=\infty$. If the $n_i$ are pairwise relatively prime then it is sufficient that $\sum \frac{1}{n_i}=\infty$.Source: erdosproblems.com/281 | Last verified: January 14, 2026