A local–global theorem on periodic maps

Let ψ1,…,ψk be maps from 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 ; moreover, there are periodic maps only depending on S such that for all . 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.