Arithmetic Properties of Periodic Maps

Let $\psi_1,\ldots,\psi_k$ be periodic maps from $\Z$ to a field of characteristic $p$ (where $p$ is zero or a prime). Assume that positive integers $n_1,\ldots,n_k$ not divisible by $p$ are their periods respectively. We show that $\psi_1+\cs+\psi_k$ is constant if $\psi_1(x)+\cs+\psi_k(x)$ equals a constant for $|S|$ consecutive integers $x$ where $S=\bigcup_{s=1}^k\{r/n_s:\ r=0,\ldots,n_s-1\}$. We also present some new results on finite systems of arithmetic sequences.

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Published 1 January 2004