Quasihomomorphisms from the integers into Hamming metrics

A function f : Z → Qn is a c-quasihomomorphism if the Hamming distance between f(x + y) and f(x) + f(y) is at most c for all x, y ∈ Z. We show that any c-quasihomomorphism has distance at most some constant C(c) to an actual group homomorphism; here C(c) depends only on c and not on n or f. This gives a positive answer to a special case of a question posed by Kazhdan and Ziegler.