Arithmetic Properties of Periodic Maps

Let ${\psi }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\psi }_k$ be periodic maps from $\mathbb{Z}$ to a field of characteristic p (where p is zero or a prime). Assume that positive integers $n_1,\mathrm{.}\mathrm{.}\mathrm{.},n_k$ not divisible by p are their periods respectively. We show that ${\psi }_1+\mathrm{.}\mathrm{.}\mathrm{.}+{\psi }_k$ is constant if ${\psi }_1(x)+\mathrm{.}\mathrm{.}\mathrm{.}+{\psi }_k(x)$ equals a constant for |S| consecutive integers x where S={r/n_s: r=0,…,n_s-1; s=1,…,k}. We also present some new results on finite systems of arithmetic sequences.