×

zbMATH — the first resource for mathematics

A survey of results on density modulo 1 of double sequences containing algebraic numbers. (English) Zbl 1231.11078
Author’s summary: “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.”
The author presents a number of theorems mainly from his recent papers.

MSC:
11J71 Distribution modulo one
37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
PDF BibTeX XML Cite
Full Text: Link EuDML
References:
[1] Akhunzhanov R.K. : On the distribution modulo 1 of exponential sequences. Math. Notes 76(2):153-160, 2004. · Zbl 1196.11107 · doi:10.1023/B:MATN.0000036753.96493.67
[2] Berend D. : Multi-invariant sets on tori. Trans. Amer. Math. Soc., 280(2):509-532, 1983. · Zbl 0532.10028 · doi:10.2307/1999631
[3] Berend D. : Multi-invariant sets on compact abelian groups. Trans. Amer. Math. Soc. 286(2):505-535, 1984. · Zbl 0523.22004 · doi:10.2307/1999808
[4] Berend D. : Actions of sets of integers on irrationals. Acta Arith. 48:275-306, 1987. · Zbl 0575.10030 · eudml:206055
[5] Berend D. : Dense \(({\mathrm mod}\,1)\) dilated semigroups of algebraic numbers. J. Number Theory, 26(3):246-256, 1987. · Zbl 0623.10038 · doi:10.1016/0022-314X(87)90082-5
[6] Boshernitzan M. D. : Elementary proof of Furstenberg’s Diophantine result. Proc. Amer. Math. Soc. 122(1):67-70, 1994. · Zbl 0815.11036 · doi:10.2307/2160842
[7] Bourgain J., Lindenstrauss E., Michel P., Venkatesh A. : Some effective results for \(\times a\) \(\times b.\). Preprint. Available online at · Zbl 1237.37009 · cims.nyu.edu
[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. · Zbl 1220.11131 · doi:10.2298/PIM0591067D · www.emis.de
[9] Furstenberg H. : Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Systems Theory, 1:1-49, 1967. · Zbl 0146.28502 · doi:10.1007/BF01692494
[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. · Zbl 1119.37008
[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. · Zbl 1050.37012 · doi:10.1017/S0143385703000440
[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.
[13] Hewitt E., Ross K. A. : Abstract harmonic analysis. volume 1. Springer, Berlin, 1994. · Zbl 0837.43002
[14] Kra B. : A generalization of Furstenberg’s Diophantine theorem. Proc. Amer. Math. Soc. 127(7):1951-1956, 1999. · Zbl 0921.11034 · doi:10.1090/S0002-9939-99-04742-5
[15] Lindenstrauss E. : Rigidity of multiparameter actions. Israel J. Math. 149:199-227, 2005. · Zbl 1155.37301 · doi:10.1007/BF02772541 · arxiv:math/0402165
[16] Meiri D. : Entropy and uniform distribution of orbits in \(\mathbb T^d\). Israel J. Math. 105:155-183, 1998. · Zbl 0908.11032 · doi:10.1007/BF02780327
[17] Meiri D., Peres Y. : Bi-invariant sets and measures have integer Hausdorff dimension. Ergodic Theory Dynam. Systems 19(2):523-534, 1999. · Zbl 0954.37004 · doi:10.1017/S014338579912100X
[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. · Zbl 1071.37008 · doi:10.1007/s10711-003-0816-x
[20] Urban R. : On density modulo \(1\) of some expressions containing algebraic integers. Acta Arith., 127(3):217-229, 2007. · Zbl 1202.11041 · doi:10.4064/aa141-2-5
[21] Urban R. : Sequences of algebraic integers and density modulo \(1\). J. Théor. Nombres Bordeaux 19(3):755-762, 2007. · Zbl 1157.11030 · doi:10.5802/jtnb.610 · eudml:55150
[22] Urban R. : Algebraic numbers and density modulo \(1\). J. Number Theory, 128(3):645-661, 2008. · Zbl 1195.11091 · doi:10.1016/j.jnt.2009.05.012
[23] Urban R.: Sequences of algebraic numbers and density modulo \(1\). Publ. Math. Debrecen, 72(1-2):141-154, 2008. · Zbl 1164.11027
[24] Urban R.: On density modulo \(1\) of some expressions containing algebraic numbers. submitted. · Zbl 1118.11034 · doi:10.4064/aa127-3-2
[25] Veech W. A. : Topological dynamics. Bull. Amer. Math. Soc. 83(5):775-830, 1977. · Zbl 0384.28018 · doi:10.1090/S0002-9904-1977-14319-X
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.