  Let $\psi_1,...,\psi_k$ be periodic maps from $\Bbb Z$ to a field of
characteristic p (where p is zero or a prime). Assume that positive integers
$n_1,...,n_k$ not divisible by p are their periods respectively. We show that
$\psi_1+...+\psi_k$ is constant if $\psi_1(x)+...+\psi_k(x)$ 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.
