A survey of results on density modulo $1$ of double sequences containing algebraic numbers

Urban, Roman

Communications in Mathematics, Tome 16 (2008) no. 1, pp. 31-43

Abstract (VO)

In this survey article we start from the famous Furstenberg theorem on non-lacunary semigroups of integers, and next we present its generalizations and some related results.

Détail

Classification : 11-02, 11J71, 37A45, 54H15, 54H20
Keywords: Algebraic numbers; density modulo $1$; uniformly distributed sequences; topological dynamics; semigroups of endomorphisms; ID-semigroup; invariant sets; $a$-adic solenoids

Bibliographie

[1] Akhunzhanov R.K. : On the distribution modulo 1 of exponential sequences. Math. Notes 76(2):153–160, 2004. | DOI | MR | Zbl

[2] Berend D. : Multi-invariant sets on tori. Trans. Amer. Math. Soc., 280(2):509–532, 1983. | DOI | MR | Zbl

[3] Berend D. : Multi-invariant sets on compact abelian groups. Trans. Amer. Math. Soc. 286(2):505–535, 1984. | DOI | MR | Zbl

[4] Berend D. : Actions of sets of integers on irrationals. Acta Arith. 48:275–306, 1987. | MR | Zbl

[5] Berend D. : Dense $({\rm mod}\,1)$ dilated semigroups of algebraic numbers. J. Number Theory, 26(3):246–256, 1987. | DOI | MR

[6] Boshernitzan M. D. : Elementary proof of Furstenberg’s Diophantine result. Proc. Amer. Math. Soc. 122(1):67–70, 1994. | MR | Zbl

[7] Bourgain J., Lindenstrauss E., Michel P., Venkatesh A. : Some effective results for $\times a$ $\times b.$. Preprint. Available online at http://cims.nyu.edu/~venkatesh/research/blmv.pdf | MR

[8] Dubickas A. : Two exercises concerning the degree of the product of algebraic numbers. Publ. Inst. Math. (Beograd) (N.S.) 77(91):67–70, 2005. | DOI | MR | Zbl

[9] Furstenberg H. : Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory, 1:1–49, 1967. | DOI | MR | Zbl

[10] Guivarc’h Y. : Renewal theorems, products of random matrices, and toral endomorphisms. Potential theory in Matsue, 53–66, Adv. Stud. Pure Math., 44, Math. Soc. Japan, Tokyo, 2006. | MR

[11] Guivarc’h Y., Starkov A. N. : Orbits of linear group actions, random walk on homogeneous spaces, and toral automorphisms. Ergodic Theory Dynam. Systems, 24(3):767–802, 2004. | MR

[12] Guivarc’h Y., Urban R. : Semigroup actions on tori and stationary measures on projective spaces. Studia Math. 171(1):33–66, 2005. Corrigendum to the paper: Studia Math. 183(2):195–196, 2007. | MR

[13] Hewitt E., Ross K. A. : Abstract harmonic analysis. volume 1. Springer, Berlin, 1994. | Zbl

[14] Kra B. : A generalization of Furstenberg’s Diophantine theorem. Proc. Amer. Math. Soc. 127(7):1951–1956, 1999. | MR

[15] Lindenstrauss E. : Rigidity of multiparameter actions. Israel J. Math. 149:199–227, 2005. | DOI | MR | Zbl

[16] Meiri D. : Entropy and uniform distribution of orbits in $\mathbb T^d$. Israel J. Math. 105:155–183, 1998. | MR

[17] Meiri D., Peres Y. : Bi-invariant sets and measures have integer Hausdorff dimension. Ergodic Theory Dynam. Systems 19(2):523–534, 1999. | MR

[18] Muchnik R. : Orbits of Zariski dense semigroups of $\mathrm {SL}(n,\mathbb Z)$. preprint

[19] Muchnik R. : Semigroup actions on $\mathbb T^n$. Geometriae Dedicata, 110:1–47, 2005. | MR

[20] Urban R. : On density modulo $1$ of some expressions containing algebraic integers. Acta Arith., 127(3):217–229, 2007. | DOI | MR | Zbl

[21] Urban R. : Sequences of algebraic integers and density modulo $1$. J. Théor. Nombres Bordeaux 19(3):755–762, 2007. | DOI | MR

[22] Urban R. : Algebraic numbers and density modulo $1$. J. Number Theory, 128(3):645–661, 2008. | DOI | MR

[23] Urban R.: Sequences of algebraic numbers and density modulo $1$. Publ. Math. Debrecen, 72(1–2):141–154, 2008. | MR | Zbl

[24] Urban R.: On density modulo $1$ of some expressions containing algebraic numbers. submitted.

[25] Veech W. A. : Topological dynamics. Bull. Amer. Math. Soc. 83(5):775–830, 1977. | DOI | MR | Zbl