On the Greatest Common Divisor of Shifted Sets

Given a set of $n$ positive integers $\{a_1, \ldots, a_n\}$ and an integer parameter $H$ we study small additive shifts of its elements by integers $h_i$ with $|h_i| \le H$, $i =1, \ldots, n$, such that the greatest common divisor of $a_1+h_1, \ldots, a_n+h_n$ is very different from that of $a_1, \ldots, a_n$. We also consider a similar problem for the least common multiple.