Skip to main content
SpringerLink
Log in

Topics in Homogeneous Dynamics and Number Theory

  • Chapter
Elements of Dynamical Systems

Part of the book series: Texts and Readings in Mathematics ((TRIM,volume 79))

Abstract

This is a survey of some topics at the interface of dynamical systems and number theory, based on lectures delivered at CIRM Luminy, the University of Houston, and IIT Delhi. Specifically, we will be interested in the ergodic theory of group actions on homogeneous spaces and its connections to metric Diophantine approximation. The topics covered in the lectures included the study of the Diophantine approximation of linear forms using dynamics, the study of quadratic forms in particular the famous Oppenheim’s conjecture and its variations, as well as lattice point counting using dynamics. At IIT, non-divergence estimates for unipotent flows and Margulis’ proof of the Borel Harish-Chandra theorem using the non-divergence estimates were also covered.

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

Access this chapter

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 79.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    This notation, mainstream in the Kleinian groups literature, is at odds with the notation in previous sections where G was the ambient Lie group and \(\Gamma \) a lattice in G.

References

  1. Alam, M., & Ghosh, A. (2020). Equidistribution on homogeneous spaces and the distribution of approximates in Diophantine approximation. Transactions of the American Mathematical Society, 373(5), 3357–3374.

    Article  MathSciNet  MATH  Google Scholar 

  2. An, J., Ghosh, A., Guan, L., & Ly, T. (2019). Bounded orbits of diagonalizable flows on finite volume quotients of products of SL(2, $\mathbb{R}$). Advances in Mathematics,354, 106743, 18 pp.

    Google Scholar 

  3. Athreya, J. S., & Ghosh, A. (2018). The Erdős-Szüsz-Turán distribution for equivariant processes. L’Enseignement Mathématique, 64(2), 1–21.

    MathSciNet  MATH  Google Scholar 

  4. Athreya, J. S., Ghosh, A., & Prasad, A. (2012). Ultrametric logarithm laws II. Monatshefte für Mathematik, 167(3), 333–356.

    Article  MathSciNet  MATH  Google Scholar 

  5. Athreya, J. S., Ghosh, A., & Tseng, J. (2014). Spherical averages of Siegel transforms for higher rank diagonal actions and applications. arXiv:1407.3573.

    Google Scholar 

  6. Athreya, J. S., Ghosh, A., & Tseng, J. (2015). Spherical averages of Siegel transforms and spiraling of lattice approximations. Journal of the London Mathematical Society, 91(2), 383–404.

    Article  MathSciNet  MATH  Google Scholar 

  7. Athreya, J. S., Biswas, K., & Ghosh, A. (2016). Diophantine approximation and lattice actions on the Clifford plane. In L. Ji, A. Papadopoulos & W. Su (Eds.), Teichmüller theory and its impact.

    Google Scholar 

  8. Badziahin, D., Pollington, A., & Velani, S. (2011). On a problem in simultaneous Diophantine approximation: Schmidt’s conjecture. Annals of Mathematics, 174(3), 1837–1883.

    Article  MathSciNet  MATH  Google Scholar 

  9. Beresnevich, V., Ramírez, F., & Velani, S. (2016). Metric Diophantine approximation: Aspects of recent work. Dynamics and analytic number theory. London mathematical society lecture note series (Vol. 437, pp. 1–95). Cambridge: Cambridge University Press.

    Google Scholar 

  10. Beresnevich, V., Ghosh, A., Simmons, D., & Velani, S. (2018). Diophantine approximation in Kleinian groups: Singular, extremal, and bad limit points. Journal of the London Mathematical Society, 98(2), 306–328.

    Article  MathSciNet  MATH  Google Scholar 

  11. Bilyk, D., Ma, X., Pipher, J., & Spencer, C. (2016). Diophantine approximation and directional discrepancy of rotated lattices. Transactions of AMS, 368, 3871–3897.

    Article  MathSciNet  MATH  Google Scholar 

  12. Borel, A. (1995). Values of indefinite quadratic forms at integral points and flows on spaces of lattices. Bulletin of the American Mathematical Society, 32, 184–204.

    Article  MathSciNet  MATH  Google Scholar 

  13. Bourgain, J. (2016). A quantitative Oppenheim theorem for generic diagonal quadratic forms. Israel Journal of Mathematics, 215(1), 503–512.

    Article  MathSciNet  MATH  Google Scholar 

  14. Broderick, R., Fishman, L., Kleinbock, D., Reich, A., & Weiss, B. (2012). The set of badly approximable vectors is strongly $C^1$ incompressible. Mathematical Proceedings of the Cambridge Philosophical Society,153(2), 319–339.

    Google Scholar 

  15. Burger, E. (1992). Homogeneous Diophantine approximation in S-integers. Pacific Journal of Mathematics, 152(2), 211–253.

    Article  MathSciNet  MATH  Google Scholar 

  16. Cassels, J. (1957). An introduction to Diophantine approximation. Cambridge tracts in mathematics and mathematical physics (Vol. 45). Cambridge: Cambridge University Press.

    Google Scholar 

  17. Choi, K. (2000). On the distribution of points in projective space of bounded height. Transactions of the American Mathematical Society, 352(3), 1071–1111.

    Article  MathSciNet  MATH  Google Scholar 

  18. Choi, K., & Vaaler, J. D. (1999). Diophantine approximation in projective space. Number theory (Ottawa, ON, 1996). CRM proceedings and lecture notes (Vol. 19, pp. 55–65). Providence: American Mathematical Society.

    Google Scholar 

  19. Chow, S., Ghosh, A., Guan, L., Marnat, A., & Simmons, D. (2018). Diophantine transference inequalities: Weighted, inhomogeneous, and intermediate exponents. Annali della Scuola Normale Superiore di Pisa. arXiv:1808.07184.

    Google Scholar 

  20. Dani, S. G. (1985). Divergent trajectories of flows on homogeneous spaces and Diophantine approximation. Journal für die Reine und Angewandte Mathematik, 359, 55–89.

    MathSciNet  MATH  Google Scholar 

  21. Dani, S. G. (2020). Article in this volume.

    Google Scholar 

  22. Davenport, H., & Schmidt, W. M. (1970). Dirichlet’s theorem on Diophantine approximation: II. Acta Arithmetica, 16, 413–424.

    Article  MathSciNet  MATH  Google Scholar 

  23. Einsiedler, M., & Ward, T. (2011). Ergodic theory with a view towards number theory. Graduate texts in mathematics (Vol. 259, pp. xviii+481). London: Springer.

    Google Scholar 

  24. Einsiedler, M., & Ward, T. (2016). Diophantine problems and homogeneous dynamics. Dynamics and analytic number theory. London mathematical society lecture note series (Vol. 437, pp. 258–288). Cambridge: Cambridge University Press.

    Google Scholar 

  25. Einsiedler, M., Ghosh, A., & Lytle, B. (2016). Badly approximable vectors, C1 curves and number fields. Ergodic Theory and Dynamical Systems, 36(6), 1851–1864.

    Article  MathSciNet  MATH  Google Scholar 

  26. Erdős, P., Szüsz, O., & Turán, P. (1958). Remarks on the theory of Diophantine approximation. Colloquium Mathematicum, 6, 119–126.

    Article  MathSciNet  MATH  Google Scholar 

  27. Eskin, A. (2010). Unipotent flows and applications. Homogeneous flows, moduli spaces and arithmetic. Clay mathematics proceedings (Vol. 10, pp. 71–129). Providence: American Mathematical Society.

    Google Scholar 

  28. Fishman, L., Simmons, D., & Urbański, M. (2018). Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces. Memoirs of the American Mathematical Society,254(1215), v+137.

    Google Scholar 

  29. Ganguly, A., & Ghosh, A. (2017). Dirichlet’s theorem in function fields. Canadian Journal of Mathematics, 69, 532–547.

    Article  MathSciNet  MATH  Google Scholar 

  30. Ghosh, A. (2018). Diophantine approximation on subspaces of n and dynamics on homogeneous spaces. Handbook of group actions. Vol. IV. Advanced lectures in mathematics (ALM) (Vol. 41, pp. 509–527). Somerville: International Press.

    Google Scholar 

  31. Ghosh, A., & Haynes, A. (2016). Projective metric number theory. Journal für die Reine und Angewandte Mathematik, 712, 39–50.

    MathSciNet  MATH  Google Scholar 

  32. Ghosh, A., & Kelmer, D. (2017). Shrinking targets for semisimple groups. Bulletin of the London Mathematical Society, 49(2), 235–245.

    Article  MathSciNet  MATH  Google Scholar 

  33. Ghosh, A., & Kelmer, D. (2018). A quantitative Oppenheim theorem for generic ternary quadratic forms. Journal of Modern Dynamics, 12, 1–8.

    Article  MathSciNet  MATH  Google Scholar 

  34. Ghosh, A., & Royals, R. (2015). An extension of the Khintchine-Groshev theorem. Acta Arithmetica, 167, 1–17.

    Article  MathSciNet  MATH  Google Scholar 

  35. Ghosh, A., Gorodnik, A., & Nevo, A. (2013). Diophantine approximation and automorphic spectrum. International Mathematics Research Notices, 21, 5002–5058.

    Article  MathSciNet  MATH  Google Scholar 

  36. Ghosh, A., Gorodnik, A., & Nevo, A. (2014). Metric Diophantine approximation on homogeneous varieties. Compositio Mathematica, 150(8), 1435–1456.

    Article  MathSciNet  MATH  Google Scholar 

  37. Ghosh, A., Gorodnik, A., & Nevo, A. (2015). Diophantine approximation exponents on homogeneous varieties. Contemporary Mathematics, 631, 181–200.

    Article  MathSciNet  MATH  Google Scholar 

  38. Ghosh, A., Gorodnik, A., & Nevo, A. (2018). Best possible rates of distribution of dense lattice orbits in homogeneous spaces. Journal für die Reine und Angewandte Mathematik, 745, 155–188.

    Article  MathSciNet  MATH  Google Scholar 

  39. Ghosh, A., Gorodnik, A., & Nevo, A. (2018). Optimal density for values of generic polynomial maps. American Journal of Mathematics. arXiv:1801.01027.

    Google Scholar 

  40. Harrap, S., & Hussain, M. (2017). A note on badly approximable sets in projective space. Mathematische Zeitschrift, 285, 239–250.

    Article  MathSciNet  MATH  Google Scholar 

  41. Hattori, T. (2007). Some Diophantine approximation inequalities and products of hyperbolic spaces. Journal of the Mathematical Society of Japan, 59, 239–264.

    Article  MathSciNet  MATH  Google Scholar 

  42. Hill, R., & Velani, S. (1995). The ergodic theory of shrinking targets. Inventiones Mathematicae, 119, 175–198.

    Article  MathSciNet  MATH  Google Scholar 

  43. Hines, R. (2017). Badly approximable numbers over imaginary quadratic fields. arXiv:1707.07231.

    Google Scholar 

  44. Hines, R. (2018). Examples of badly approximable vectors over number fields. arXiv:1809.07404.

    Google Scholar 

  45. Jarnik, V. (1929). Diophantischen approximationen und Hausdorffsches mass. Matematicheskii Sbornik, 36, 371–382.

    MATH  Google Scholar 

  46. Kesten, H. (1962). Some probabilistic theorems on Diophantine approximations. Transactions of the American Mathematical Society, 103, 189–217.

    Article  MathSciNet  MATH  Google Scholar 

  47. Kesten, H., & Sós, V. (1966). On two problems of Erdös, Szüsz and Turán concerning Diophantine approximations. Acta Arithmetica, 12, 183–192.

    Article  MathSciNet  MATH  Google Scholar 

  48. Khintchine, A. (1926). Über eine klasse linearer Diophantische approximationen. Rendiconti del Circolo Matematico di Palermo, 50, 170–195.

    Article  MATH  Google Scholar 

  49. Kleinbock, D. (2001). Some applications of homogeneous dynamics to number theory. Smooth ergodic theory and its applications (Seattle, WA, 1999). Proceedings of symposia in pure mathematics (Vol. 69, pp. 639–660). Providence: American Mathematical Society.

    Google Scholar 

  50. Kleinbock, D. (2010). Quantitative nondivergence and its Diophantine applications. Homogeneous flows, moduli spaces and arithmetic. Clay mathematics proceedings (Vol. 10, pp. 131–153). Providence: American Mathematical Society.

    Google Scholar 

  51. Kleinbock, D., & Ly, T. (2016). Badly approximable $S$-numbers and absolute Schmidt games. Journal of Number Theory,164, 13–42.

    Google Scholar 

  52. Kleinbock, D., & Margulis, G. A. (1999). Logarithm laws for flows on homogeneous spaces. Inventiones Mathematicae, 138, 451–494.

    Article  MathSciNet  MATH  Google Scholar 

  53. Kleinbock, D., Shah, N., & Starkov, A. (2002). Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory. Handbook of dynamical systems (Vol. 1A, pp. 813–930). Amsterdam: North-Holland.

    Google Scholar 

  54. Kleinbock, D., Shi, R., & Weiss, B. (2017). Pointwise equidistribution with an error rate and with respect to unbounded functions. Mathematische Annalen, 367(1–2), 857–879.

    Article  MathSciNet  MATH  Google Scholar 

  55. Mahler, K. (1961). Lectures on Diophantine approximations (pp. 181–188). Notre Dame: University of Notre Dame.

    Google Scholar 

  56. Margulis, G. (1987). Formes quadratriques indéfinies et flots unipotents sur les espaces homogènes. Comptes Rendus de l’Académie des Sciences Paris Séries I - Mathematics,304(10), 249–253.

    Google Scholar 

  57. Marklof, J. (2000). The $n$-point correlations between values of a linear form. Ergodic Theory and Dynamical Systems,20(4), 1127–1172.

    Google Scholar 

  58. Maucourant, F. (2009). Arithmetical and geometrical aspects of homogeneous Diophantine approximation by algebraic numbers in a given number field. Dynamical systems and Diophantine approximation. Séminaires and congres (Vol. 19, pp. 37–48). Paris: Société Mathématique de France.

    Google Scholar 

  59. Morris, D. W. (2005). Ratner’s theorems on unipotent flows. Chicago lectures in mathematics. Chicago: University of Chicago Press.

    Google Scholar 

  60. Oh, H. (2010). Orbital counting via mixing and unipotent flows. Homogeneous flows, moduli spaces and arithmetic. Clay mathematics proceedings (Vol. 10, pp. 339–375). Providence: American Mathematical Society.

    Google Scholar 

  61. Patterson, S. J. (1976). The limit set of a Fuchsian group. Acta Mathematica, 136, 241–273.

    Article  MathSciNet  MATH  Google Scholar 

  62. Patterson, S. J. (1976). Diophantine approximation in Fuchsian groups. Philosophical Transactions of the Royal Society London Series A, 282, 527–563.

    Article  MathSciNet  MATH  Google Scholar 

  63. Quéme, R. (1992). On Diophantine approximation by algebraic numbers of a given number field: A new generalization of Dirichlet approximation theorem. Journées Arithmétiques (1989) (Luminy, 1989), Astérisque,198–200 (1991), 273–283.

    Google Scholar 

  64. Rumely, R. S. (1989). Capacity theory on algebraic curves. Lecture notes in mathematics (Vol. 1378). New York: Springer.

    Google Scholar 

  65. Schmidt, W. M. (1966). On badly approximable numbers and certain games. Transactions of the American Mathematical Society, 123, 178–199.

    Article  MathSciNet  MATH  Google Scholar 

  66. Schmidt, W. M. (1975). Simultaneous approximation to algebraic numbers by elements of a number field. Monatshefte für Mathematik, 79, 55–66.

    Article  MathSciNet  MATH  Google Scholar 

  67. Schmidt, W. M. (1980). Diophantine approximation. Lecture notes in mathematics (Vol. 785). Berlin: Springer.

    Google Scholar 

  68. Sullivan, D. (1983). Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics. Acta Mathematica, 144, 215–237.

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This survey grow out of lectures delivered at CIRM Luminy, at the University of Houston and at IIT Delhi. I am grateful to the organisers of each of the three events for inviting me and for their hospitality. Special thanks to Jayadev Athreya, Alan Haynes and Riddhi Shah. I would also like to thank the editors of this volume, Anima Nagar, Riddhi Shah and Shrihari Sridharan. This work was supported by a grant from the Indo-French Centre for the Promotion of Advanced Research; a Department of Science and Technology, Government of India Swarnajayanti fellowship; a MATRICS grant from the Science and Engineering Research Board; and the Benoziyo Endowment Fund for the Advancement of Science at the Weizmann Institute. I gratefully acknowledge the hospitality of the Technion and the Weizmann Institute.

Author information

Authors and Affiliations

  1. School of Mathematics, Tata Institute for Fundamental Research, Mumbai, India

    Anish Ghosh

Authors
  1. Anish Ghosh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anish Ghosh .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, India

    Anima Nagar

  2. School of Physical Sciences, Jawaharlal Nehru University, New Delhi, India

    Riddhi Shah

  3. School of Mathematics, Indian Institute of Science Education and Research - Thiruvananthapuram, Thiruvananthapuram, India

    Shrihari Sridharan

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Hindustan Book Agency

About this chapter

Cite this chapter

Ghosh, A. (2020). Topics in Homogeneous Dynamics and Number Theory. In: Nagar, A., Shah, R., Sridharan, S. (eds) Elements of Dynamical Systems. Texts and Readings in Mathematics, vol 79. Springer, Singapore. https://doi.org/10.1007/978-981-16-7962-9_6

Download citation

Publish with us

Policies and ethics

Search

Navigation