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).