Contractions and expansion

Let A@?R be a finite set and let K=1 be a real number. Suppose that for each a@?A we are given an injective map @f"a:A-R which fixes a and contracts other points towards it in the sense that |a-@f"a(x)|==K10|A|-O"K(1). An immediate consequence of this is the estimate |A+K@?A|=K10|A|-O"K(1), which is a slightly weakened version of a result of Bukh.