SOLVED - $100
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).
