Encyclopedia of Complexity and Systems Science

2009 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Ergodic Theory on Homogeneous Spaces and Metric Number Theory

  • Dmitry Kleinbock
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30440-3_180

Definition of the Subject

The theory of Diophantine approximation , named after Diophantus of Alexandria, in its simplest set-up deals with the approximation of real numbersby rational numbers. Various higher‐dimensional generalizations involve studying values of linear or polynomial maps at integer points. Oftena certain “approximation property” is fixed, and one wants to characterize the set of numbers (vectors, matrices) which share thisproperty, by means of certain measures (Lebesgue, or Hausdorff, or some other interesting measures). This is usually referred to as metric Diophantine approximation .

The starting point for the theory is an elementary fact that ℚ, the set of rational numbers, is dense in ℝ, the reals. In other words,every real number can be approximated by rationals: for any \( { y\in\mathbb{R} } \)
This is a preview of subscription content, log in to check access.

Notes

Acknowledgment

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

Bibliography

  1. 1.
    Baker A (1975) Transcendental number theory. Cambridge University Press,LondonzbMATHGoogle Scholar
  2. 2.
    Baker RC (1976) Metric diophantine approximation on manifolds. J Lond Math Soc14:43–48ADSzbMATHGoogle Scholar
  3. 3.
    Baker RC (1978) Dirichlet's theorem on diophantine approximation. Math ProcCambridge Phil Soc 83:37–59zbMATHGoogle Scholar
  4. 4.
    Bekka M, Mayer M (2000) Ergodic theory and topological dynamics of group actionson homogeneous spaces. Cambridge University Press, CambridgezbMATHGoogle Scholar
  5. 5.
    Beresnevich V (1999) On approximation of real numbers by real algebraic numbers.Acta Arith 90:97–112MathSciNetzbMATHGoogle Scholar
  6. 6.
    Beresnevich V (2002) A Groshev type theorem for convergence on manifolds. ActaMathematica Hungarica 94:99–130MathSciNetzbMATHGoogle Scholar
  7. 7.
    Beresnevich V, Bernik VI, Kleinbock D, Margulis GA (2002) Metric Diophantineapproximation: the Khintchine–Groshev theorem for nondegenerate manifolds. Moscow Math J 2:203–225MathSciNetzbMATHGoogle Scholar
  8. 8.
    Beresnevich V, Dickinson H, Velani S (2006) Measure theoretic laws for lim supsets. Mem Amer Math Soc:179:1–91Google Scholar
  9. 9.
    Beresnevich V, Dickinson H, Velani S (2007) Diophantine approximation on planarcurves and the distribution of rational points. Ann Math 166:367–426MathSciNetzbMATHGoogle Scholar
  10. 10.
    Beresnevich V, Velani S (2007) A note on simultaneous Diophantineapproximation on planar curves. Ann Math 337:769–796MathSciNetzbMATHGoogle Scholar
  11. 11.
    Bernik VI (1984) A proof of Baker's conjecture in the metric theory oftranscendental numbers. Dokl Akad Nauk SSSR 277:1036–1039MathSciNetGoogle Scholar
  12. 12.
    Bernik VI, Dodson MM (1999) Metric Diophantine approximation on manifolds.Cambridge University Press, CambridgezbMATHGoogle Scholar
  13. 13.
    Bernik VI, Kleinbock D, Margulis GA (2001) Khintchine-type theorems onmanifolds: convergence case for standard and multiplicative versions. Int Math Res Notices 2001:453–486MathSciNetzbMATHGoogle Scholar
  14. 14.
    Besicovitch AS (1929) On linear sets of points of fractional dimensions. AnnMath 101:161–193MathSciNetzbMATHGoogle Scholar
  15. 15.
    Bugeaud Y (2002) Approximation by algebraic integers and Hausdorff dimension.J London Math Soc 65:547–559MathSciNetzbMATHGoogle Scholar
  16. 16.
    Bugeaud Y (2004) Approximation by algebraic numbers. Cambridge UniversityPress, CambridgezbMATHGoogle Scholar
  17. 17.
    Cassels JWS (1957) An introduction to Diophantine approximation. CambridgeUniversity Press, New YorkzbMATHGoogle Scholar
  18. 18.
    Cassels JWS, Swinnerton-Dyer H (1955) On the product of three homogeneouslinear forms and the indefinite ternary quadratic forms. Philos Trans Roy Soc London Ser A 248:73–96MathSciNetADSzbMATHGoogle Scholar
  19. 19.
    Dani SG (1979) On invariant measures, minimal sets and a lemma of Margulis.Invent Math 51:239–260MathSciNetADSzbMATHGoogle Scholar
  20. 20.
    Dani SG (1985) Divergent trajectories of flows on homogeneous spaces anddiophantine approximation. J Reine Angew Math 359:55–89MathSciNetzbMATHGoogle Scholar
  21. 21.
    Dani SG (1986) On orbits of unipotent flows on homogeneous spaces. II.Ergodic Theory Dynam Systems 6:167–182MathSciNetzbMATHGoogle Scholar
  22. 22.
    Dani SG, Margulis GA (1993) Limit distributions of orbits of unipotent flowsand values of quadratic forms. In: IM Gelfand Seminar. American Mathematical Society, Providence,pp 91–137Google Scholar
  23. 23.
    Davenport H, Schmidt WM (1970) Dirichlet's theorem on diophantineapproximation. In: Symposia Mathematica. INDAM, Rome, pp 113–132Google Scholar
  24. 24.
    Davenport H, Schmidt WM (1969/1970) Dirichlet's theorem on diophantineapproximation. II. Acta Arith 16:413–424Google Scholar
  25. 25.
    Ding J (1994) A proof of a conjecture of C. L. Siegel. J Number Theory46:1–11MathSciNetzbMATHGoogle Scholar
  26. 26.
    Dodson MM (1992) Hausdorff dimension, lower order and Khintchine's theorem inmetric Diophantine approximation. J Reine Angew Math 432:69–76MathSciNetzbMATHGoogle Scholar
  27. 27.
    Dodson MM (1993) Geometric and probabilistic ideas in the metric theory ofDiophantine approximations. Uspekhi Mat Nauk 48:77–106MathSciNetzbMATHGoogle Scholar
  28. 28.
    Druţu C (2005) Diophantine approximation on rational quadrics. Ann Math333:405–469Google Scholar
  29. 29.
    Duffin RJ, Schaeffer AC (1941) Khintchine's problem in metric Diophantineapproximation. Duke Math J 8:243–255MathSciNetGoogle Scholar
  30. 30.
    Einsiedler M, Katok A, Lindenstrauss E (2006) Invariant measures and the setof exceptions to Littlewood's conjecture. Ann Math 164:513–560MathSciNetzbMATHGoogle Scholar
  31. 31.
    Einsiedler M, Kleinbock D (2007) Measure rigidity and p‑adic Littlewood-type problems. Compositio Math 143:689–702MathSciNetzbMATHGoogle Scholar
  32. 32.
    Einsiedler M, Lindenstrauss E (2006) Diagonalizable flows on locallyhomogeneous spaces and number theory. In: Proceedings of the International Congress of Mathematicians. Eur Math Soc, Zürich,pp 1731–1759Google Scholar
  33. 33.
    Eskin A (1998) Counting problems and semisimple groups. In: Proceedings ofthe International Congress of Mathematicians. Doc Math, Berlin, pp 539–552Google Scholar
  34. 34.
    Eskin A, Margulis GA, Mozes S (1998) Upper bounds and asymptoticsin a quantitative version of the Oppenheim conjecture. Ann Math 147:93–141MathSciNetzbMATHGoogle Scholar
  35. 35.
    Eskin A, Margulis GA, Mozes S (2005) Quadratic forms of signature (2,2) andeigenvalue spacings on rectangular 2-tori. Ann Math 161:679–725MathSciNetzbMATHGoogle Scholar
  36. 36.
    Fishman L (2006) Schmidt's games on certain fractals. Israel S Math (toappear)Google Scholar
  37. 37.
    Gallagher P (1962) Metric simultaneous diophantine approximation. J LondonMath Soc 37:387–390MathSciNetzbMATHGoogle Scholar
  38. 38.
    Ghosh A (2005) A Khintchine type theorem for hyperplanes. J London MathSoc 72:293–304MathSciNetzbMATHGoogle Scholar
  39. 39.
    Ghosh A (2007) Metric Diophantine approximation over a local field of positivecharacteristic. J Number Theor 124:454–469zbMATHGoogle Scholar
  40. 40.
    Ghosh A (2006) Dynamics on homogeneous spaces and Diophantine approximation onmanifolds. Ph D Thesis, Brandeis University, WalthumGoogle Scholar
  41. 41.
    Gorodnik A (2007) Open problems in dynamics and related fields. J Mod Dyn1:1–35MathSciNetzbMATHGoogle Scholar
  42. 42.
    Groshev AV (1938) Une théorème sur les systèmes des formes linéaires. DoklAkad Nauk SSSR 9:151–152Google Scholar
  43. 43.
    Harman G (1998) Metric number theory. Clarendon Press, Oxford UniversityPress, New YorkGoogle Scholar
  44. 44.
    Hutchinson JE (1981) Fractals and self-similarity. Indiana Univ Math J30:713–747MathSciNetzbMATHGoogle Scholar
  45. 45.
    Jarnik V (1928-9) Zur metrischen Theorie der diophantischen Approximationen.Prace Mat-Fiz 36:91–106Google Scholar
  46. 46.
    Jarnik V (1929) Diophantischen Approximationen und Hausdorffsches Mass. MatSb 36:371–382zbMATHGoogle Scholar
  47. 47.
    Khanin K, Lopes-Dias L, Marklof J (2007) Multidimensional continued fractions,dynamical renormalization and KAM theory. Comm Math Phys 270:197–231MathSciNetADSzbMATHGoogle Scholar
  48. 48.
    Khintchine A (1924) Einige Sätze über Kettenbrüche, mit Anwendungen auf dieTheorie der Diophantischen Approximationen. Math Ann 92:115–125MathSciNetzbMATHGoogle Scholar
  49. 49.
    Khintchine A (1963) Continued fractions. P Noordhoff Ltd,GroningenzbMATHGoogle Scholar
  50. 50.
    Kleinbock D (2001) Some applications of homogeneous dynamics to number theory.In: Smooth ergodic theory and its applications. American Mathematical Society, Providence, pp 639–660Google Scholar
  51. 51.
    Kleinbock D (2003) Extremal subspaces and their submanifolds. Geom Funct Anal13:437–466MathSciNetzbMATHGoogle Scholar
  52. 52.
    Kleinbock D (2004) Baker–Sprindžuk conjectures for complex analyticmanifolds. In: Algebraic groups and Arithmetic. Tata Inst Fund Res, Mumbai, pp 539–553Google Scholar
  53. 53.
    Kleinbock D (2008) An extension of quantitative nondivergence and applicationsto Diophantine exponents. Trans AMS, to appearGoogle Scholar
  54. 54.
    Kleinbock D, Lindenstrauss E, Weiss B (2004) On fractal measures anddiophantine approximation. Selecta Math 10:479–523MathSciNetzbMATHGoogle Scholar
  55. 55.
    Kleinbock D, Margulis GA (1996) Bounded orbits of nonquasiunipotent flows onhomogeneous spaces. In: Sinaĭ's Moscow Seminar on Dynamical Systems. American Mathematical Society, Providence,pp 141–172Google Scholar
  56. 56.
    Kleinbock D, Margulis GA (1998) Flows on homogeneous spaces and Diophantineapproximation on manifolds. Ann Math 148:339–360MathSciNetzbMATHGoogle Scholar
  57. 57.
    Kleinbock D, Margulis GA (1999) Logarithm laws for flows on homogeneousspaces. Invent Math 138:451–494MathSciNetADSzbMATHGoogle Scholar
  58. 58.
    Kleinbock D, Shah N, Starkov A (2002) Dynamics of subgroup actions onhomogeneous spaces of Lie groups and applications to number theory. In: Handbook on Dynamical Systems, vol 1A. Elsevier Science, North Holland,pp 813–930Google Scholar
  59. 59.
    Kleinbock D, Tomanov G (2007) Flows on S‑arithmetic homogeneous spaces and applications to metric Diophantine approximation. Comm Math Helv82:519–581MathSciNetzbMATHGoogle Scholar
  60. 60.
    Kleinbock D, Weiss B (2005) Badly approximable vectors on fractals. Israel JMath 149:137–170MathSciNetzbMATHGoogle Scholar
  61. 61.
    KleinbockD, Weiss B (2005) Friendly measures, homogeneous flows and singularvectors. In: Algebraic and Topological Dynamics. American Mathematical Society, Providence,pp 281–292Google Scholar
  62. 62.
    Kleinbock D, Weiss B (2008) Dirichlet's theorem on diophantine approximationand homogeneous flows. J Mod Dyn 2:43–62MathSciNetzbMATHGoogle Scholar
  63. 63.
    Kontsevich M, Suhov Y (1999) Statistics of Klein polyhedra andmultidimensional continued fractions. In: Pseudoperiodic topology. American Mathematical Society, Providence,pp 9–27Google Scholar
  64. 64.
    Kristensen S, Thorn R, Velani S (2006) Diophantine approximation and badlyapproximable sets. Adv Math 203:132–169MathSciNetzbMATHGoogle Scholar
  65. 65.
    Lagarias JC (1994) Geodesic multidimensional continued fractions. Proc LondonMath Soc 69:464–488MathSciNetzbMATHGoogle Scholar
  66. 66.
    Lindenstrauss E (2007) Some examples how to use measure classification innumber theory. In: Equidistribution in number theory, an introduction. Springer, Dordrecht, pp 261–303Google Scholar
  67. 67.
    Mahler K (1932) Über das Mass der Menge aller S-Zahlen. Math Ann106:131–139MathSciNetGoogle Scholar
  68. 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 365–370Google Scholar
  69. 69.
    Margulis GA (1989) Discrete subgroups and ergodic theory. In: Number theory,trace formulas and discrete groups (Oslo, 1987). Academic Press, Boston, pp 377–398Google Scholar
  70. 70.
    Margulis GA (1997) Oppenheim conjecture. In: Fields Medalists' lectures.World Sci Publishing, River Edge, pp 272–327Google Scholar
  71. 71.
    Margulis GA (2002) Diophantine approximation, lattices and flows onhomogeneous spaces. In: A panorama of number theory or the view from Baker's garden. Cambridge University Press, Cambridge,pp 280–310Google Scholar
  72. 72.
    MauldinD, Urbański M (1996) Dimensions and measures in infinite iteratedfunction systems. Proc London Math Soc 73:105–154Google Scholar
  73. 73.
    Mohammadi A, Salehi Golsefidy A (2008) S‑Arithmetic Khintchine-Type Theorem. PreprintGoogle Scholar
  74. 74.
    Moore CC (1966) Ergodicity of flows on homogeneous spaces. Amer J Math88:154–178MathSciNetzbMATHGoogle Scholar
  75. 75.
    Roth KF (1955) Rational Approximations to Algebraic Numbers. Mathematika2:1–20MathSciNetGoogle Scholar
  76. 76.
    Schmidt WM (1960) A metrical theorem in diophantine approximation. Canad JMath 12:619–631MathSciNetzbMATHGoogle Scholar
  77. 77.
    Schmidt WM (1964) Metrische Sätze über simultane Approximation abhängigerGrößen. Monatsh Math 63:154–166Google Scholar
  78. 78.
    Schmidt WM (1969) Badly approximable systems of linear forms. J Number Theory1:139–154MathSciNetADSzbMATHGoogle Scholar
  79. 79.
    Schmidt WM (1972) Norm form equations. Ann Math 96:526–551zbMATHGoogle Scholar
  80. 80.
    Schmidt WM (1980) Diophantine approximation. Springer,BerlinzbMATHGoogle Scholar
  81. 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 239–254Google Scholar
  82. 82.
    Sprindžuk VG (1964) More on Mahler's conjecture. Dokl Akad Nauk SSSR155:54–56Google Scholar
  83. 83.
    Sprindžuk VG (1969) Mahler's problem in metric number theory. AmericanMathematical Society, ProvidenceGoogle Scholar
  84. 84.
    Sprindžuk VG (1979) Metric theory of Diophantine approximations. VHWinston & Sons, Washington DCGoogle Scholar
  85. 85.
    Sprindžuk VG (1980) Achievements and problems of the theory ofDiophantine approximations. Uspekhi Mat Nauk 35:3–68Google Scholar
  86. 86.
    Starkov A (2000) Dynamical systems on homogeneous spaces. AmericanMathematical Society, ProvidenceGoogle Scholar
  87. 87.
    Sullivan D (1982) Disjointspheres, approximation by imaginary quadraticnumbers, and the logarithm law for geodesics. Acta Math 149:215–237MathSciNetzbMATHGoogle Scholar
  88. 88.
    Stratmann B, Urbanski M (2006)Diophantine extremality of the Pattersonmeasure. Math Proc Cambridge Phil Soc 140:297–304MathSciNetzbMATHGoogle Scholar
  89. 89.
    Urbanski M (2005) Diophantine approximation of self-conformal measures. J Number Theory 110:219–235MathSciNetzbMATHGoogle Scholar
  90. 90.
    Želudevič F (1986) Simultane diophantische Approximationenabhängiger Grössen in mehreren Metriken. Acta Arith 46:285–296Google Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Dmitry Kleinbock
    • 1
  1. 1.Department of MathematicsBrandeis UniversityWalthamUSA