A local-global theorem on periodic maps

Let ψ1,…,ψk be maps from Z to an additive abelian group with positive periods n1,…,nk, respectively. We show that the function ψ=ψ1+⋯+ψk is constant if ψ(x) equals a constant for |S| consecutive integers x where S={r/ns:r=0,…,ns−1;s=1,…,k}; moreover, there are periodic maps f0,…,f|S|−1:Z→Z only depending on S such that ψ(x)=∑r=0|S|−1fr(x)ψ(r) for all x∈Z. This local-global theorem extends a previous result [Z.W. Sun, Arithmetic properties of periodic maps, Math. Res. Lett. 11 (2004) 187-196], and has various applications.