Straightening Theorem for bounded Abelian groups

We prove that every continuous function f with f(0)=0 between two bounded Abelian groups G and H equipped with the Bohr topology coincides with a homomorphism when restricted to an infinite subset of the domain. This extends the main results of [K. Kunen, Bohr topology and partition theorems for vector spaces, Topology Appl. 90 (1998) 97–107, D. Dikranjan, S. Watson, A solution to van Douwen's problem on the Bohr topologies, J. Pure Appl. Algebra 163 (2001) 147–158]. Moreover, we give several applications and we answer a question of [B. Givens, K. Kunen, Chromatic numbers and Bohr topologies, Topology Appl. 131 (2) (2003) 189–202].