Sequences of algebraic integers and density modulo 1
type: Item
instance of: scholarly article
title: Sequences of algebraic integers and density modulo 1
zbMATH Open document ID: 1157.11030
author: Roman Urban
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2008-07-21T00:00:00Z
full work available at URL: https://eudml.org/doc/55150
review text: Let \(\lambda,\mu\) be two algebraic integers such that all their powers \(\lambda^k,\mu^\ell\quad (k,\ell\geq 1)\) are of degree \(d\geq 2\) and distinct. Suppose further that \(\lambda,\mu\) have modulus greater than \(1\), and that \(\mu\) can be written as a polynomial in \(\lambda\) having integer coefficients, and that for the conjugates \(\lambda_i\neq \lambda\) and \(\mu_i\neq \mu\quad(i=2,\dots,d)\) of \(\lambda,\mu\) we have \(|\lambda_i|\neq 1\) and \(\max_i(|\lambda_i|,|\mu_i|)>1\). Then it is proved that for any \(\xi\neq 0\) and sequence of real numbers \(r_m\) the sequence \(\{\mu^m\lambda^n\xi+r_m:n,m\in\mathbb N\}\) is dense modulo \(1\). Earlier Kra had proved the corresponding result in the case \(d=1\), except that \(\xi\) was necessarily restricted to being irrational. The author suspects that the assumption that \(\mu\) is a polynomial in \(\lambda\) with integer coefficients is not necessary. Reviewer's remark: On the other hand, the condition \(\max_i(|\lambda_i|,|\mu_i|)>1\) cannot be removed. Otherwise one could take \(d=2\), \(\lambda=\frac{3+\sqrt{5}}{2}\), \(\mu=2\lambda\), \(\lambda_2=1/\lambda<1\), \(\mu_2=2/\lambda<1\), \(\xi=1\) and \(r_m=0\) for all \(m\). Then since \(\mu^m\lambda^n+ \mu_2^m\lambda_2^n\) is an integer, the only limit point of the sequence \(\{\mu^m\lambda^n\xi+r_m\}\) in this case is the point \(1\).
Mathematics Subject Classification ID: 11J71
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 37A25
zbMATH DE Number: 5302622
zbMATH Keywords: Density modulo 1
zbMATH Keywords: algebraic integers
zbMATH Keywords: topological dynamics
zbMATH Keywords: ID-semigroups
MaRDI profile type: Publication
OpenAlex ID: W2328298301
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DOI: 10.5802/JTNB.610
description: scientific article; zbMATH DE number 5302622