OPEN
Let $A=\{n_1<\cdots<n_r\}$ be a finite set of integers. What is the maximum density of integers covered by a suitable choice of congruences $a_i\pmod{n_i}$?
Is the minimum density achieved when all the $a_i$ are equal?
Simpson
[Si86] has observed that the density of integers covered is at least
\[\sum_i \frac{1}{n_i}-\sum_{i<j}\frac{1}{[n_i,n_j]}+\sum_{i<j<k}\frac{1}{[n_i,n_j,n_k]}-\cdot\]
(where $[\cdots]$ denotes the least common multiple) which is achieved when all $a_i$ are equal, settling the second question.
