A Local - Global Theorem on Periodic Maps 3

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) = |S|−1 r=0 fr(x)ψ(r) for all x ∈ Z. This local-global theorem extends a previous result [Math. Res. Lett. 11(2004), 187–196], and has various applications.