Skip to main content
SpringerLink
Log in
Book cover

Mathematics of Complexity and Dynamical Systems pp 302–312Cite as

Ergodic Theory on Homogeneous Spaces and Metric Number Theory

  • Reference work entry

Article Outline

Glossary

Definition of the Subject

Introduction

Basic Facts

Connection with Dynamics on the Space of Lattices

Diophantine Approximation with Dependent Quantities: The Set-Up

Further Results

Future Directions

Acknowledgment

Bibliography

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   600.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Abbreviations

Diophantine approximation :

Diophantine approximation refers to approximation of real numbers by rational numbers, or more generally, finding integer points at which some (possibly vector‐valued) functions attain values close to integers.

Metric number theory :

Metric number theory (or, specifically, metric Diophantine approximation) refers to the study of sets of real numbers or vectors with prescribed Diophantine approximation properties.

Homogeneous spaces:

A homogeneous space \({G/\Gamma}\) of a group G by its subgroup Γ is the space of cosets \({\{g\Gamma\}}\). When G is a Lie group and Γ is a discrete subgroup, the space \({G/\Gamma}\) is a smooth manifold and locally looks like G itself.

Lattice; unimodular lattice :

A lattice in a Lie group is a discrete subgroup of finite covolume; unimodular stands for covolume equal to 1.

Ergodic theory :

The study of statistical properties of orbits in abstract models of dynamical systems.

Hausdorff dimension :

A nonnegative number attached to a metric space and extending the notion of topological dimension of “sufficiently regular” sets, such as smooth submanifolds of real Euclidean spaces.

Bibliography

  1. Baker A (1975) Transcendental number theory. Cambridge University Press, London

    Book  MATH  Google Scholar 

  2. Baker RC (1976) Metric diophantine approximation on manifolds. J Lond Math Soc 14:43–48

    Article  MATH  Google Scholar 

  3. Baker RC (1978) Dirichlet's theorem on diophantine approximation. Math Proc Cambridge Phil Soc 83:37–59

    Article  MATH  Google Scholar 

  4. Bekka M, Mayer M (2000) Ergodic theory and topological dynamics of group actions on homogeneous spaces. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  5. Beresnevich V (1999) On approximation of real numbers by real algebraic numbers. Acta Arith 90:97–112

    MathSciNet  MATH  Google Scholar 

  6. Beresnevich V (2002) A Groshev type theorem for convergence on manifolds. Acta Mathematica Hungarica 94:99–130

    Article  MathSciNet  MATH  Google Scholar 

  7. Beresnevich V, Bernik VI, Kleinbock D, Margulis GA (2002) Metric Diophantine approximation: the Khintchine–Groshev theorem for nondegenerate manifolds. Moscow Math J 2:203–225

    MathSciNet  MATH  Google Scholar 

  8. Beresnevich V, Dickinson H, Velani S (2006) Measure theoretic laws for lim sup sets. Mem Amer Math Soc:179:1–91

    Google Scholar 

  9. Beresnevich V, Dickinson H, Velani S (2007) Diophantine approximation on planar curves and the distribution of rational points. Ann Math 166:367–426

    Article  MathSciNet  MATH  Google Scholar 

  10. Beresnevich V, Velani S (2007) A note on simultaneous Diophantine approximation on planar curves. Ann Math 337:769–796

    Article  MathSciNet  MATH  Google Scholar 

  11. Bernik VI (1984) A proof of Baker's conjecture in the metric theory of transcendental numbers. Dokl Akad Nauk SSSR 277:1036–1039

    MathSciNet  Google Scholar 

  12. Bernik VI, Dodson MM (1999) Metric Diophantine approximation on manifolds. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  13. Bernik VI, Kleinbock D, Margulis GA (2001) Khintchine-type theorems on manifolds: convergence case for standard and multiplicative versions. Int Math Res Notices 2001:453–486

    Article  MathSciNet  MATH  Google Scholar 

  14. Besicovitch AS (1929) On linear sets of points of fractional dimensions. Ann Math 101:161–193

    Article  MathSciNet  MATH  Google Scholar 

  15. Bugeaud Y (2002) Approximation by algebraic integers and Hausdorff dimension. J London Math Soc 65:547–559

    Article  MathSciNet  MATH  Google Scholar 

  16. Bugeaud Y (2004) Approximation by algebraic numbers. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  17. Cassels JWS (1957) An introduction to Diophantine approximation. Cambridge University Press, New York

    MATH  Google Scholar 

  18. Cassels JWS, Swinnerton-Dyer H (1955) On the product of three homogeneous linear forms and the indefinite ternary quadratic forms. Philos Trans Roy Soc London Ser A 248:73–96

    Article  MathSciNet  MATH  Google Scholar 

  19. Dani SG (1979) On invariant measures, minimal sets and a lemma of Margulis. Invent Math 51:239–260

    Article  MathSciNet  MATH  Google Scholar 

  20. Dani SG (1985) Divergent trajectories of flows on homogeneous spaces and diophantine approximation. J Reine Angew Math 359:55–89

    MathSciNet  MATH  Google Scholar 

  21. Dani SG (1986) On orbits of unipotent flows on homogeneous spaces. II. Ergodic Theory Dynam Systems 6:167–182

    MathSciNet  MATH  Google Scholar 

  22. Dani SG, Margulis GA (1993) Limit distributions of orbits of unipotent flows and values of quadratic forms. In: IM Gelfand Seminar. American Mathematical Society, Providence, pp 91–137

    Google Scholar 

  23. Davenport H, Schmidt WM (1970) Dirichlet's theorem on diophantine approximation. In: Symposia Mathematica. INDAM, Rome, pp 113–132

    Google Scholar 

  24. Davenport H, Schmidt WM (1969/1970) Dirichlet's theorem on diophantine approximation. II. Acta Arith 16:413–424

    Google Scholar 

  25. Ding J (1994) A proof of a conjecture of C. L. Siegel. J Number Theory 46:1–11

    Article  MathSciNet  MATH  Google Scholar 

  26. Dodson MM (1992) Hausdorff dimension, lower order and Khintchine's theorem in metric Diophantine approximation. J Reine Angew Math 432:69–76

    MathSciNet  MATH  Google Scholar 

  27. Dodson MM (1993) Geometric and probabilistic ideas in the metric theory of Diophantine approximations. Uspekhi Mat Nauk 48:77–106

    MathSciNet  MATH  Google Scholar 

  28. Druţu C (2005) Diophantine approximation on rational quadrics. Ann Math 333:405–469

    Google Scholar 

  29. Duffin RJ, Schaeffer AC (1941) Khintchine's problem in metric Diophantine approximation. Duke Math J 8:243–255

    Article  MathSciNet  Google Scholar 

  30. Einsiedler M, Katok A, Lindenstrauss E (2006) Invariant measures and the set of exceptions to Littlewood's conjecture. Ann Math 164:513–560

    Article  MathSciNet  MATH  Google Scholar 

  31. Einsiedler M, Kleinbock D (2007) Measure rigidity and p‑adic Littlewood-type problems. Compositio Math 143:689–702

    MathSciNet  MATH  Google Scholar 

  32. Einsiedler M, Lindenstrauss E (2006) Diagonalizable flows on locally homogeneous spaces and number theory. In: Proceedings of the International Congress of Mathematicians. Eur Math Soc, Zürich, pp 1731–1759

    Google Scholar 

  33. Eskin A (1998) Counting problems and semisimple groups. In: Proceedings of the International Congress of Mathematicians. Doc Math, Berlin, pp 539–552

    Google Scholar 

  34. Eskin A, Margulis GA, Mozes S (1998) Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture. Ann Math 147:93–141

    Article  MathSciNet  MATH  Google Scholar 

  35. Eskin A, Margulis GA, Mozes S (2005) Quadratic forms of signature (2,2) and eigenvalue spacings on rectangular 2-tori. Ann Math 161:679–725

    Article  MathSciNet  MATH  Google Scholar 

  36. Fishman L (2006) Schmidt's games on certain fractals. Israel S Math (to appear)

    Google Scholar 

  37. Gallagher P (1962) Metric simultaneous diophantine approximation. J London Math Soc 37:387–390

    Article  MathSciNet  MATH  Google Scholar 

  38. Ghosh A (2005) A Khintchine type theorem for hyperplanes. J London Math Soc 72:293–304

    Article  MathSciNet  MATH  Google Scholar 

  39. Ghosh A (2007) Metric Diophantine approximation over a local field of positive characteristic. J Number Theor 124:454–469

    Article  MATH  Google Scholar 

  40. Ghosh A (2006) Dynamics on homogeneous spaces and Diophantine approximation on manifolds. Ph D Thesis, Brandeis University, Walthum

    Google Scholar 

  41. Gorodnik A (2007) Open problems in dynamics and related fields. J Mod Dyn 1:1–35

    Article  MathSciNet  MATH  Google Scholar 

  42. Groshev AV (1938) Une théorème sur les systèmes des formes linéaires. Dokl Akad Nauk SSSR 9:151–152

    Google Scholar 

  43. Harman G (1998) Metric number theory. Clarendon Press, Oxford University Press, New York

    Google Scholar 

  44. Hutchinson JE (1981) Fractals and self-similarity. Indiana Univ Math J 30:713–747

    Article  MathSciNet  MATH  Google Scholar 

  45. Jarnik V (1928-9) Zur metrischen Theorie der diophantischen Approximationen. Prace Mat-Fiz 36:91–106

    Google Scholar 

  46. Jarnik V (1929) Diophantischen Approximationen und Hausdorffsches Mass. Mat Sb 36:371–382

    MATH  Google Scholar 

  47. Khanin K, Lopes-Dias L, Marklof J (2007) Multidimensional continued fractions, dynamical renormalization and KAM theory. Comm Math Phys 270:197–231

    Article  MathSciNet  MATH  Google Scholar 

  48. Khintchine A (1924) Einige Sätze über Kettenbrüche, mit Anwendungen auf die Theorie der Diophantischen Approximationen. Math Ann 92:115–125

    Article  MathSciNet  MATH  Google Scholar 

  49. Khintchine A (1963) Continued fractions. P Noordhoff Ltd, Groningen

    MATH  Google Scholar 

  50. Kleinbock D (2001) Some applications of homogeneous dynamics to number theory. In: Smooth ergodic theory and its applications. American Mathematical Society, Providence, pp 639–660

    Chapter  Google Scholar 

  51. Kleinbock D (2003) Extremal subspaces and their submanifolds. Geom Funct Anal 13:437–466

    Article  MathSciNet  MATH  Google Scholar 

  52. Kleinbock D (2004) Baker–Sprindžuk conjectures for complex analytic manifolds. In: Algebraic groups and Arithmetic. Tata Inst Fund Res, Mumbai, pp 539–553

    Google Scholar 

  53. Kleinbock D (2008) An extension of quantitative nondivergence and applications to Diophantine exponents. Trans AMS, to appear

    Google Scholar 

  54. Kleinbock D, Lindenstrauss E, Weiss B (2004) On fractal measures and diophantine approximation. Selecta Math 10:479–523

    MathSciNet  MATH  Google Scholar 

  55. Kleinbock D, Margulis GA (1996) Bounded orbits of nonquasiunipotent flows on homogeneous spaces. In: Sinaĭ's Moscow Seminar on Dynamical Systems. American Mathematical Society, Providence, pp 141–172

    Google Scholar 

  56. Kleinbock D, Margulis GA (1998) Flows on homogeneous spaces and Diophantine approximation on manifolds. Ann Math 148:339–360

    Article  MathSciNet  MATH  Google Scholar 

  57. Kleinbock D, Margulis GA (1999) Logarithm laws for flows on homogeneous spaces. Invent Math 138:451–494

    Article  MathSciNet  MATH  Google Scholar 

  58. Kleinbock D, Shah N, Starkov A (2002) Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory. In: Handbook on Dynamical Systems, vol 1A. Elsevier Science, North Holland, pp 813–930

    Chapter  Google Scholar 

  59. Kleinbock D, Tomanov G (2007) Flows on S‑arithmetic homogeneous spaces and applications to metric Diophantine approximation. Comm Math Helv 82:519–581

    Article  MathSciNet  MATH  Google Scholar 

  60. Kleinbock D, Weiss B (2005) Badly approximable vectors on fractals. Israel J Math 149:137–170

    Article  MathSciNet  MATH  Google Scholar 

  61. Kleinbock D, Weiss B (2005) Friendly measures, homogeneous flows and singular vectors. In: Algebraic and Topological Dynamics. American Mathematical Society, Providence, pp 281–292

    Google Scholar 

  62. Kleinbock D, Weiss B (2008) Dirichlet's theorem on diophantine approximation and homogeneous flows. J Mod Dyn 2:43–62

    MathSciNet  MATH  Google Scholar 

  63. Kontsevich M, Suhov Y (1999) Statistics of Klein polyhedra and multidimensional continued fractions. In: Pseudoperiodic topology. American Mathematical Society, Providence, pp 9–27

    Google Scholar 

  64. Kristensen S, Thorn R, Velani S (2006) Diophantine approximation and badly approximable sets. Adv Math 203:132–169

    Article  MathSciNet  MATH  Google Scholar 

  65. Lagarias JC (1994) Geodesic multidimensional continued fractions. Proc London Math Soc 69:464–488

    Article  MathSciNet  MATH  Google Scholar 

  66. Lindenstrauss E (2007) Some examples how to use measure classification in number theory. In: Equidistribution in number theory, an introduction. Springer, Dordrecht, pp 261–303

    Google Scholar 

  67. Mahler K (1932) Über das Mass der Menge aller S-Zahlen. Math Ann 106:131–139

    Article  MathSciNet  Google Scholar 

  68. Margulis GA (1975) On the action of unipotent groups in the space of lattices. In: Lie groups and their representations (Budapest, 1971). Halsted, New York, pp 365–370

    Google Scholar 

  69. Margulis GA (1989) Discrete subgroups and ergodic theory. In: Number theory, trace formulas and discrete groups (Oslo, 1987). Academic Press, Boston, pp 377–398

    Google Scholar 

  70. Margulis GA (1997) Oppenheim conjecture. In: Fields Medalists' lectures. World Sci Publishing, River Edge, pp 272–327

    Chapter  Google Scholar 

  71. Margulis GA (2002) Diophantine approximation, lattices and flows on homogeneous spaces. In: A panorama of number theory or the view from Baker's garden. Cambridge University Press, Cambridge, pp 280–310

    Google Scholar 

  72. Mauldin D, Urbański M (1996) Dimensions and measures in infinite iterated function systems. Proc London Math Soc 73:105–154

    Google Scholar 

  73. Mohammadi A, Salehi Golsefidy A (2008) S‑Arithmetic Khintchine-Type Theorem. Preprint

    Google Scholar 

  74. Moore CC (1966) Ergodicity of flows on homogeneous spaces. Amer J Math 88:154–178

    Article  MathSciNet  MATH  Google Scholar 

  75. Roth KF (1955) Rational Approximations to Algebraic Numbers. Mathematika 2:1–20

    Article  MathSciNet  Google Scholar 

  76. Schmidt WM (1960) A metrical theorem in diophantine approximation. Canad J Math 12:619–631

    Article  MathSciNet  MATH  Google Scholar 

  77. Schmidt WM (1964) Metrische Sätze über simultane Approximation abhängiger Größen. Monatsh Math 63:154–166

    Article  Google Scholar 

  78. Schmidt WM (1969) Badly approximable systems of linear forms. J Number Theory 1:139–154

    Article  MathSciNet  MATH  Google Scholar 

  79. Schmidt WM (1972) Norm form equations. Ann Math 96:526–551

    Article  MATH  Google Scholar 

  80. Schmidt WM (1980) Diophantine approximation. Springer, Berlin

    MATH  Google Scholar 

  81. Sheingorn M (1993) Continued fractions and congruence subgroup geodesics. In: Number theory with an emphasis on the Markoff spectrum (Provo, UT, 1991). Dekker, New York, pp 239–254

    Google Scholar 

  82. Sprindžuk VG (1964) More on Mahler's conjecture. Dokl Akad Nauk SSSR 155:54–56

    Google Scholar 

  83. Sprindžuk VG (1969) Mahler's problem in metric number theory. American Mathematical Society, Providence

    Google Scholar 

  84. Sprindžuk VG (1979) Metric theory of Diophantine approximations. VH Winston & Sons, Washington DC

    Google Scholar 

  85. Sprindžuk VG (1980) Achievements and problems of the theory of Diophantine approximations. Uspekhi Mat Nauk 35:3–68

    Google Scholar 

  86. Starkov A (2000) Dynamical systems on homogeneous spaces. American Mathematical Society, Providence

    Google Scholar 

  87. Sullivan D (1982) Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics. Acta Math 149:215–237

    Article  MathSciNet  MATH  Google Scholar 

  88. Stratmann B, Urbanski M (2006) Diophantine extremality of the Patterson measure. Math Proc Cambridge Phil Soc 140:297–304

    Article  MathSciNet  MATH  Google Scholar 

  89. Urbanski M (2005) Diophantine approximation of self-conformal measures. J Number Theory 110:219–235

    Article  MathSciNet  MATH  Google Scholar 

  90. Želudevič F (1986) Simultane diophantische Approximationen abhängiger Grössen in mehreren Metriken. Acta Arith 46:285–296

    Google Scholar 

Download references

Acknowledgment

The work on this paper was supported in part by NSF Grant DMS-0239463.

Author information

Authors and Affiliations

  1. Department of Mathematics, Brandeis University, Waltham, USA

    Dmitry Kleinbock

Authors
  1. Dmitry Kleinbock
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag

About this entry

Cite this entry

Kleinbock, D. (2012). Ergodic Theory on Homogeneous Spaces and Metric Number Theory. In: Meyers, R. (eds) Mathematics of Complexity and Dynamical Systems. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1806-1_19

Download citation

Publish with us

Policies and ethics

Search

Navigation