Incomplete character sums and a special class of permutations (Q1826280)

Let \(G\) be a finite abelian group and let \(\psi\) be a permutation of \(G\). The authors study properties of the sequence defined by \(u_{n+1}=\psi(u_n)\) for some initial value \(u_0\). As \(G\) is finite, the sequence is periodic with some least period \(t\). For segments of length \(N<t\) and \(\chi\) a character of \(G\) they investigate the incomplete character sum \[ \sum_{n=0}^{N-1}\chi(u_n). \] The study involves the complete character sum \(\sum_{g\in G}\chi\left(\psi^r(g)-g\right)\) for some integer \(r\). This is particularly easy if \(\psi^r- \text{id}\) is a permutation of \(G\) as then the value is zero. This leads to the definition of \(\mathcal{R}\)-orthomorphisms, namely \(\psi\) is an \(\mathcal{R}\)-orthomorphism if \(\psi^r- \text{id}\) is a permutation for all \(r\in \mathcal{R}\). For such permutations \(\psi\) better bounds on the incomplete character sum are proven.