Arithmetic Properties of Periodic Maps

Abstract

Let ${\psi }_1,...,{\psi }_k$ be periodic maps from $\mathbb{Z}$ to a field of characteristic p (where p is zero or a prime). Assume that positive integers $n_1,...,n_k$ 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.

10 pages; accepted by Math. Res. Lett. Also available from http://pweb.nju.edu.cn/zwsun