Recurrent sequences and endomorphisms of Euclidean spaces (Q1337833)

Let \(k\) be a natural number, \(\zeta_ 1,\dots, \zeta_ k\) complex numbers of modulus 1 and let \(g: \mathbb{N}\to \mathbb{C}\) be a function defined by \[ g(n)= c_ 1 \zeta_ 1^ n+ \cdots+ c_ k \zeta_ k^ n\tag \(*\) \] with coefficients \(c_ 1,\dots, c_ k\in \mathbb{C}\). P. Enflo posed the problem to determine the smallest number \(N= N(k)\) such that for every \(g\) of the form \((*)\) \(| g(1)|= \cdots= | g(N)|\) implies \(| g(n)|= | g(1)|\) for all \(n\in \mathbb{N}\). Using the fact that every \(g\) of the form \((*)\) solves a linear recurrence relation the authors give the estimate \(N(k)\leq k^ 2- k+1\), which is sharp for infinitely many \(k\in \mathbb{N}\). In order to show this they employ a result of J. Singer on perfect difference sets. From this it follows that for every prime power \(k\neq 1\) there exists a \(q\)- periodic function \(g\) of the form \((*)\) with \[ | g(1)|= \cdots= | g(q-1) |\neq| g(q)|, \qquad q=k^ 2- k+1. \] This disproves two conjectures of Enflo concerning the order of \(N(k)\) resp. of \(q\). Further the authors solve a related problem of P. Enflo on endomorphisms of the Euclidean space \(\mathbb{C}^ k\). They refer to a paper of \textit{P. Enflo}, \textit{V. I. Gurarii}, \textit{V. Lomonosov} and \textit{Yu. I. Lyubich} [Linear Algebra Appl. (to appear)] dealing with the same topic.