Let $n_1<n_2<\cdots$ be an infinite sequence of integers with associated $a_k\pmod{n_k}$, such that for some $\epsilon>0$ we have $n_k>(1+\epsilon)k\log k$ for all $k$. Then $$\#\{ m<n_k : m\not\equiv a_i\pmod{n_i} \textrm{ for }1\leq i\leq k\}\neq o(k).$$

Let $n_1<n_2<\cdots$ be an infinite sequence of integers with associated $a_k\pmod{n_k}$, such that for some $\epsilon>0$ we have $n_k>(1+\epsilon)k\log k$ for all $k$. Then $$\#\{ m<n_k : m\not\equiv a_i\pmod{n_i} \textrm{ for }1\leq i\leq k\}\neq o(k).$$

Cambie observed that this is false.