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An $\epsilon$-almost covering system is a set of congruences $a_i\pmod{n_i}$ for distinct moduli $n_1<\ldots<n_k$ such that the density of those integers which satisfy none of them is $\leq \epsilon$. Is there a constant $C>1$ such that for every $\epsilon>0$ and $N\geq 1$ there is an $\epsilon$-almost covering system with $N\leq n_1<\cdots <n_k\leq CN$?
By a simple averaging argument the set of moduli $[m_1,m_2]\cap \mathbb{N}$ has a choice of residue classes which form an $\epsilon(m_1,m_2)$-almost covering system with \[\epsilon(m_1,m_2)=\prod_{m_1\leq m\leq m_2}(1-1/m).\] A $0$-covering system is just a covering system, and so by Hough [Ho15] these only exist for $n_1<10^{18}$.

The answer is no, as proved by Filaseta, Ford, Konyagin, Pomerance, and Yu [FFKPY07], who (among other results) prove that if \[1< C \leq N^{\frac{\log\log\log N}{4\log\log N}}\] then, for any $N\leq n_1<\cdots< n_k\leq CN$, the density of integers not covered for any fixed choice of residue classes is at least \[\prod_{i}(1-1/n_i)\] (and this density is achieved for some choice of residue classes as above).

Additional thanks to: Mehtaab Sawhney