On the Greatest Common Divisor of Shifted Sets

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