A local-global theorem on periodic maps

Let ${\psi }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\psi }_k$ be maps from Z to an additive abelian group with positive periods $n_1,\mathrm{.}\mathrm{.}\mathrm{.},n_k$ respectively. We show that the function $\psi ={\psi }_1+\mathrm{.}\mathrm{.}\mathrm{.}+{\psi }_k$ 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 $f_0,\mathrm{.}\mathrm{.}\mathrm{.},f_{\mathrm{\mid }S\mathrm{\mid }-1}$ from Z to Z only depending on S such that $\psi (x)={\sum }_{r=0}^{\mathrm{\mid }S\mathrm{\mid }-1}{f}_r(x)\psi (r)$ for all integers x. This local-global theorem extends a previous result [Math. Res. Lett. 11(2004), 187–196], and has various applications.