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 ϕ a : A → R which fixes a and contracts other points towards it in the sense that | a − ϕ a ( x ) | ⩽ 1 K | a − x | for all x ∈ A , and such that ϕ a ( x ) always lies between a and x . Then | ⋃ a ∈ A ϕ a ( A ) | ⩾ K 10 | A | − O K ( 1 ) . An immediate consequence of this is the estimate | A + K ⋅ A | ⩾ K 10 | A | − O K ( 1 ) , which is a slightly weakened version of a result of Bukh.