On the greatest common divisor of shifted sets

Given a set of n positive integers {a1,…,an} and an integer parameter H we study the greatest common divisor of small additive shifts of its elements by integers hi with |hi|≤H, i=1,…,n. In particular, we show that for any choice of a1,…,an there are shifts of this type for which the greatest common divisor of a1+h1,…,an+hn is much larger than H.