Sequences of algebraic integers and density modulo 1
Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 3, pp. 755-762.

We prove density modulo 1 of the sets of the form

{μmλnξ+rm:n,m},

where λ,μ is a pair of rationally independent algebraic integers of degree d2, satisfying some additional assumptions, ξ0, and r m is any sequence of real numbers.

Nous établissons la densité modulo 1 des ensembles de la forme

{μmλnξ+rm:n,m},

λ,μ sont deux entiers algébriques de degré d2, qui sont rationnellement indépendants et satisfont des hypothèses techniques supplémentaires, ξ0, et r m une suite quelconque de nombres réels.

Received:
Published online:
DOI: 10.5802/jtnb.610
Keywords: Density modulo 1, algebraic integers, topological dynamics, ID-semigroups
Roman Urban 1

1 Institute of Mathematics Wroclaw University Plac Grunwaldzki 2/4 50-384 Wroclaw, Poland
@article{JTNB_2007__19_3_755_0,
     author = {Roman Urban},
     title = {Sequences of algebraic integers and density modulo~$1$},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {755--762},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {3},
     year = {2007},
     doi = {10.5802/jtnb.610},
     zbl = {1157.11030},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.610/}
}
TY  - JOUR
TI  - Sequences of algebraic integers and density modulo $1$
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2007
DA  - 2007///
SP  - 755
EP  - 762
VL  - 19
IS  - 3
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.610/
UR  - https://zbmath.org/?q=an%3A1157.11030
UR  - https://doi.org/10.5802/jtnb.610
DO  - 10.5802/jtnb.610
LA  - en
ID  - JTNB_2007__19_3_755_0
ER  - 
%0 Journal Article
%T Sequences of algebraic integers and density modulo $1$
%J Journal de Théorie des Nombres de Bordeaux
%D 2007
%P 755-762
%V 19
%N 3
%I Université Bordeaux 1
%U https://doi.org/10.5802/jtnb.610
%R 10.5802/jtnb.610
%G en
%F JTNB_2007__19_3_755_0
Roman Urban. Sequences of algebraic integers and density modulo $1$. Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 3, pp. 755-762. doi : 10.5802/jtnb.610. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.610/

[1] D. Berend, Multi-invariant sets on tori. Trans. Amer. Math. Soc. 280 (1983), no. 2, 509–532. | MR: 716835 | Zbl: 0532.10028

[2] D. Berend, Multi-invariant sets on compact abelian groups. Trans. Amer. Math. Soc. 286 (1984), no. 2, 505–535. | MR: 760973 | Zbl: 0523.22004

[3] D. Berend, Dense ( mod 1) dilated semigroups of algebraic numbers. J. Number Theory 26 (1987), no. 3, 246–256. | MR: 901238 | 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. | MR: 213508 | 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. | MR: 2060998 | 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. | MR: 2182271 | 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. | MR: 1326374 | Zbl: 0878.58020

[8] S. Kolyada and L. Snoha, Some aspects of topological transitivity – a survey. Grazer Math. Ber. 334 (1997), 3–35. | MR: 1644768 | Zbl: 0907.54036

[9] B. Kra, A generalization of Furstenberg’s Diophantine theorem. Proc. Amer. Math. Soc. 127 (1999), no. 7, 1951–1956. | MR: 1487320 | Zbl: 0921.11034

[10] D. Meiri, Entropy and uniform distribution of orbits in 𝕋 d . Israel J. Math. 105 (1998), 155–183. | MR: 1639747 | 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. | MR: 419394 | Zbl: 0281.10001

[12] R. Mañé, Ergodic theory and differentiable dynamics. Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Springer-Verlag, Berlin, 1987. | MR: 889254 | Zbl: 0616.28007

[13] R. Muchnik, Semigroup actions on 𝕋 n . Geometriae Dedicata 110 (2005), 1–47. | MR: 2136018 | 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. | MR: 1159963 | Zbl: 0758.58024

[15] R. Urban, On density modulo 1 of some expressions containing algebraic integers. Acta Arith., 127 (2007), no. 3, 217–229. | MR: 2310344 | Zbl: 1118.11034

Cited by Sources: