Recurrent sequences and endomorphisms of Euclidean spaces
type: Item
instance of: scholarly article
title: Recurrent sequences and endomorphisms of Euclidean spaces
zbMATH Open document ID: 0812.11013
DOI: 10.1007/BF01196304
author: Cordelia Methfessel
author: Lutz G. Lucht
published in: Archiv der Mathematik
publication date: 1994-11-13T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 11B37
Mathematics Subject Classification ID: 11A25
zbMATH DE Number: 687091
zbMATH Keywords: arithmetic functions
zbMATH Keywords: endomorphisms of Euclidean space
zbMATH Keywords: Enflo conjectures
zbMATH Keywords: linear recurrence
zbMATH Keywords: perfect difference sets
reviewed by: Thomas Maxsein
MaRDI profile type: Publication
cites work: Q4847816
cites work: Uniform distribution of sequences of integers in residue classes
cites work: A Theorem in Finite Projective Geometry and Some Applications to Number Theory
full work available at URL: https://doi.org/10.1007/bf01196304
OpenAlex ID: W1966079486
description: scientific article; zbMATH DE number 687091