Urban, Roman Sequences of algebraic integers and density modulo 1. (English) Zbl 1157.11030 J. Théor. Nombres Bordx. 19, No. 3, 755-762 (2007). 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 \(1\). Reviewer: Chris Smyth (Edinburgh) Cited in 1 Document MSC: 11J71 Distribution modulo one 11R04 Algebraic numbers; rings of algebraic integers 37A25 Ergodicity, mixing, rates of mixing Keywords:Density modulo 1; algebraic integers; topological dynamics; ID-semigroups PDFBibTeX XMLCite \textit{R. Urban}, J. Théor. Nombres Bordx. 19, No. 3, 755--762 (2007; Zbl 1157.11030) Full Text: DOI EuDML Link References: [1] D. Berend, Multi-invariant sets on tori. Trans. Amer. Math. Soc. 280 (1983), no. 2, 509-532. · Zbl 0532.10028 [2] D. Berend, Multi-invariant sets on compact abelian groups. Trans. Amer. Math. Soc. 286 (1984), no. 2, 505-535. · Zbl 0523.22004 [3] D. Berend, Dense \(({\rm mod}\,1)\) dilated semigroups of algebraic numbers. J. Number Theory 26 (1987), no. 3, 246-256. · Zbl 0623.10038 [4] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory 1 (1967), 1-49. · Zbl 0146.28502 [5] Y. Guivarc’h and A. N. Starkov, Orbits of linear group actions, random walk on homogeneous spaces, and toral automorphisms. Ergodic Theory Dynam. Systems 24 (2004), no. 3, 767-802. · Zbl 1050.37012 [6] Y. Guivarc’h and R. Urban, Semigroup actions on tori and stationary measures on projective spaces. Studia Math. 171 (2005), no. 1, 33-66. · Zbl 1087.37022 [7] A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems. Encyclopedia of Mathematics and its Applications 54, Cambridge University Press, Cambridge, 1995. · Zbl 0878.58020 [8] S. Kolyada and L. Snoha, Some aspects of topological transitivity - a survey. Grazer Math. Ber. 334 (1997), 3-35. · Zbl 0907.54036 [9] B. Kra, A generalization of Furstenberg’s Diophantine theorem. Proc. Amer. Math. Soc. 127 (1999), no. 7, 1951-1956. · Zbl 0921.11034 [10] D. Meiri, Entropy and uniform distribution of orbits in \(\mathbb{T}^d\). Israel J. Math. 105 (1998), 155-183. · Zbl 0908.11032 [11] L. Kuipers and H. Niederreiter, Uniform distribution of sequences. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. · Zbl 0281.10001 [12] R. Mañé, Ergodic theory and differentiable dynamics. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Springer-Verlag, Berlin, 1987. · Zbl 0616.28007 [13] R. Muchnik, Semigroup actions on \(\mathbb{T}^n\). Geometriae Dedicata 110 (2005), 1-47. · Zbl 1071.37008 [14] S. Silverman, On maps with dense orbits and the definition of chaos. Rocky Mt. J. Math. 22 (1992), no. 1, 353-375. · Zbl 0758.58024 [15] R. Urban, On density modulo \(1\) of some expressions containing algebraic integers. Acta Arith., 127 (2007), no. 3, 217-229. · Zbl 1118.11034 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.