A local-global theorem on periodic maps

Let $\psi_1,...,\psi_k$ be maps from Z to an additive abelian group with positive periods $n_1,...,n_k$ respectively. We show that the function $\psi=\psi_1+...+\psi_k$ is constant if $\psi(x)$ equals a constant for |S| consecutive integers x where S={r/n_s: r=0,...,n_s-1; s=1,...,k}; moreover, there are periodic maps $f_0,...,f_{|S|-1}$ from Z to Z only depending on S such that $\psi(x)=\sum_{r=0}^{|S|-1}f_r(x)\psi(r)$ for all integers x. This local-global theorem extends a previous result [Math. Res. Lett. 11(2004), 187--196], and has various applications.