Sequences of algebraic integers and density modulo  1

Roman Urban[1]

  • [1] Institute of Mathematics Wroclaw University Plac Grunwaldzki 2/4 50-384 Wroclaw, Poland

Journal de Théorie des Nombres de Bordeaux (2007)

  • Volume: 19, Issue: 3, page 755-762
  • ISSN: 1246-7405

Abstract

top
We prove density modulo 1 of the sets of the form { μ m λ n ξ + r m : n , m ∈ ℕ } , where λ , μ ∈ ℝ is a pair of rationally independent algebraic integers of degree d ≥ 2 , satisfying some additional assumptions, ξ ≠ 0 , and r m is any sequence of real numbers.

How to cite

top

Urban, Roman. "Sequences of algebraic integers and density modulo $1$." Journal de Théorie des Nombres de Bordeaux 19.3 (2007): 755-762. <http://eudml.org/doc/249962>.

@article{Urban2007,
abstract = {We prove density modulo $1$ of the sets of the form\begin\{equation*\} \lbrace \mu ^m\lambda ^n\xi +r\_m:n,m\in \mathbb\{N\}\rbrace , \end\{equation*\}where $\lambda ,\mu \in \mathbb\{R\}$ is a pair of rationally independent algebraic integers of degree $d\ge 2,$ satisfying some additional assumptions, $\xi \ne 0,$ and $r_m$ is any sequence of real numbers.},
affiliation = {Institute of Mathematics Wroclaw University Plac Grunwaldzki 2/4 50-384 Wroclaw, Poland},
author = {Urban, Roman},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Density modulo $1,$ algebraic integers; topological dynamics; ID-semigroups; Density modulo 1; algebraic integers},
language = {eng},
number = {3},
pages = {755-762},
publisher = {Université Bordeaux 1},
title = {Sequences of algebraic integers and density modulo $1$},
url = {http://eudml.org/doc/249962},
volume = {19},
year = {2007},
}

TY - JOUR
AU - Urban, Roman
TI - Sequences of algebraic integers and density modulo $1$
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2007
PB - Université Bordeaux 1
VL - 19
IS - 3
SP - 755
EP - 762
AB - We prove density modulo $1$ of the sets of the form\begin{equation*} \lbrace \mu ^m\lambda ^n\xi +r_m:n,m\in \mathbb{N}\rbrace , \end{equation*}where $\lambda ,\mu \in \mathbb{R}$ is a pair of rationally independent algebraic integers of degree $d\ge 2,$ satisfying some additional assumptions, $\xi \ne 0,$ and $r_m$ is any sequence of real numbers.
LA - eng
KW - Density modulo $1,$ algebraic integers; topological dynamics; ID-semigroups; Density modulo 1; algebraic integers
UR - http://eudml.org/doc/249962
ER -

References

top
  1. D. Berend, Multi-invariant sets on tori. Trans. Amer. Math. Soc. 280 (1983), no. 2, 509–532. Zbl0532.10028MR716835
  2. D. Berend, Multi-invariant sets on compact abelian groups. Trans. Amer. Math. Soc. 286 (1984), no. 2, 505–535. Zbl0523.22004MR760973
  3. D. Berend, Dense ( mod 1 ) dilated semigroups of algebraic numbers. J. Number Theory 26 (1987), no. 3, 246–256. Zbl0623.10038MR901238
  4. H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory 1 (1967), 1–49. Zbl0146.28502MR213508
  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. Zbl1050.37012MR2060998
  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. Zbl1087.37022MR2182271
  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. Zbl0878.58020MR1326374
  8. S. Kolyada and L. Snoha, Some aspects of topological transitivity – a survey. Grazer Math. Ber. 334 (1997), 3–35. Zbl0907.54036MR1644768
  9. B. Kra, A generalization of Furstenberg’s Diophantine theorem. Proc. Amer. Math. Soc. 127 (1999), no. 7, 1951–1956. Zbl0921.11034MR1487320
  10. D. Meiri, Entropy and uniform distribution of orbits in 𝕋 d . Israel J. Math. 105 (1998), 155–183. Zbl0908.11032MR1639747
  11. L. Kuipers and H. Niederreiter, Uniform distribution of sequences. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Zbl0281.10001MR419394
  12. R. Mañé, Ergodic theory and differentiable dynamics. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Springer-Verlag, Berlin, 1987. Zbl0616.28007MR889254
  13. R. Muchnik, Semigroup actions on 𝕋 n . Geometriae Dedicata 110 (2005), 1–47. Zbl1071.37008MR2136018
  14. S. Silverman, On maps with dense orbits and the definition of chaos. Rocky Mt. J. Math. 22 (1992), no. 1, 353–375. Zbl0758.58024MR1159963
  15. R. Urban, On density modulo 1 of some expressions containing algebraic integers. Acta Arith., 127 (2007), no. 3, 217–229. Zbl1118.11034MR2310344

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.